TY - JOUR
AU - Hooley, Christopher
AB - Abstract Let $$f(x_1,\ldots ,x_8)$$ be a cubic form in eight variables with rational integral coefficients and non-zero discriminant. Then, assuming a Riemann hypothesis for certain Hasse–Weil $$L$$-functions, we prove that the indeterminate equation $$f(x_1,\ldots ,x_8)=0$$ has a non-zero solution provided that the form satisfy the necessary condition that it have a non-trivial zero in every $$p$$-adic field $${\Bbb {Q}}_p$$. This extends the earlier unconditional results due to Heath–Brown and the author for cubic forms in ten and nine variables. 1. Introduction Homogeneous cubic indeterminate equations   \[f( {{{\bf{{l}}}}}) = f( {{l}_{1}, \dotsc , {l}_{{n}}}) = 0; \quad {{{\bf{{l}}}}} \neq 0\] (1) have been the subject of not a little study, the main purpose of which has been to establish solubility for values of $$n$$ down to good limits when $$f( {{{\bf{{x}}}}})$$ satisfies the obvious necessary condition that it have a non-trivial zero in every $$p$$-adic field $${\Bbb {Q}}_p$$. Being concerned in this memoir with the largely unexplored case where $$f( {{{\bf{{x}}}}})$$ is an octonary form, we therefore begin with a brief resumé of previous work and point out that the proviso of $$p$$-adic solubility becomes superfluous in much of the discussion because it is always met whenever $$n \geq 10$$. Following work by Lewis and Birch, Davenport provided values of $$n$$ in a series of papers that culminated in [2], in which he proved solubility for any form $$f( {{{\bf{{x}}}}})$$ when $$n \geq 16$$; recently, however, this value of $$n$$ has been reduced to $$14$$ by Heath-Brown through an improvement of Davenport's method [7]. But, in the meanwhile, Heath-Brown had turned his attention to non-singular forms (that is, those with non-vanishing discriminants) and, by using the Hardy–Littlewood circle method with Kloosterman refinement, showed that the value $$n=10$$ was attributable to such forms. In that paper ([4], referred to as Ha in what follows (a table giving the designations of certain references is given at the end)), in which trigonometrical sums of the type   \[Q_r ( {{{\bf{{m}}}}},k) = \sum _{\substack {0 <h \leq k \\ {({h}, {k})} = 1}} \quad \sum _{{0 <{{{\bf{{l}}}}} \leq k}} e^{2 \pi i (h f( {{{\bf{{l}}}}}) - {{{\bf{{m}}}}} {{{\bf{{l}}}}} +r \bar {h})/k}\] occur, Heath-Brown enquired whether his result remained true for $$n=9$$ and whether the condition of non-singularity could be relaxed to permit the hypersurface $$ {{\cal {{{V}}}}}$$ defined by $$f( {{{\bf{{x}}}}}) = 0$$ to have a finite number of singularities. In fact, the first question was answered by us in the affirmative through a substantial alteration in Heath-Brown's method, in which two of the ingredients were a smoothed version of the circle method and results obtained by Katz on sums of moduli of $$Q_r ( {{{\bf{{m}}}}}, k)$$ (see [9], referred to later as Ib). Then, further extending our method, we also answered the second query positively by demonstrating solubility even for $$n=9$$ when the hypersurface $$ {{\cal {{{V}}}}}$$ admitted linearly independent ordinary double points ([10], referred to later as Ic), a result we have recently improved by showing that the condition of linear independence can be deleted [14]. Sums such as $$Q_r ( {{{\bf{{m}}}}}, k)$$ almost inevitably appear in any analysis of cubic equations of type (1) through an application of the Hardy–Littlewood circle method that employs major arcs only. Their character and function depending on whether or not $$r = 0$$, we find in the former instance the sum is truly multiplicative in the sense that $$Q_0 ( {{{\bf{{m}}}}}, k^{\prime } k^{\prime \prime }) = Q_0 ( {{{\bf{{m}}}}}, k^{\prime }) Q_0 ( {{{\bf{{m}}}}}, k^{\prime \prime })$$ when $$(k^{\prime }, k^{\prime \prime }) = 1$$, whereas in the latter instance the sums occur in the Kloosterman refinement that is needed to balance the inequality in lengths of the Farey arcs centred around fractions with denominator $$k$$. To go beyond this refinement, which in the first place is simply to be regarded as the natural extension of Kloosterman's classical method, is difficult within the framework just described because the nature of $$Q_r ( {{{\bf{{m}}}}}, k)$$ for $$r \neq 0$$ may preclude additional savings through a summation over $$k$$. Yet, in our paper on sums of three cubes ([8], referred to as Ia; see also [13]), we were able to avoid the problem identified above because, the underlying function in (1) being then   \[l_1^3 +l_2^3 +l_3^3 - l_4^3 - l_5^3 - l_6^3\] to express the condition that two sums of three cubes be equal, the integrand emerging from the application of the circle method is non-negative. This meant, since we were there interested only in gaining an upper bound of appropriate magnitude, that all Farey arcs corresponding to the modulus $$k$$ could be slightly extended to be equal with the result that all the sums $$Q_r ( {{{\bf{{m}}}}}, k)$$ for non-zero $$r$$ disappeared from the work and that we were only left with terms containing $$Q_0 ( {{{\bf{{m}}}}}, k)$$. Of these, those with $$ {{{\bf{{m}}}}} = 0$$ appeared in a singular series in the usual way, while those answering to another given $$ {{{\bf{{m}}}}}$$ could be grouped together with the possibility of their being summed over $$k$$ with some cancellation. Actually, in the case in point, the proper multiplicativity of $$Q_0 ( {{{\bf{{m}}}}}, k)$$ meant that their summation could be performed efficiently provided that we were assured of the truth of a hypothesis HW about certain Hasse–Weil $$L$$-functions that were postulated, inter alia, to conform to a certain type of Riemann hypothesis. In this case, we were therefore furnished with the first example in the literature of what we termed a double Kloosterman refinement, namely, a procedure in which estimations were performed not only by taking together a set of terms corresponding to a given denominator $$k$$ but also by then considering non-trivially the collective contribution of such sets after summation over $$k$$. And we should add that it is natural to apply the designation Kloosterman refinement to anything resembling the first stage of this process, although in doing so we may be stretching the meaning of the term because Kloosterman's original technique might no longer be in play. The favourable features presented by the problem in Ia just discussed became more generally present as a consequence of a revolutionary new method introduced by Heath-Brown in 1996 ([5], referred to as Hb). Although it was termed by him a new form of the circle method, its genesis does not lie in an integral over the unit interval but in a special function that was defined by Duke, Friedlander, and Iwaniec. Nevertheless, when applied to problems concerning equations such as (1) with $$f( {{{\bf{{x}}}}})$$ either quadratic or cubic, the new method after a difficult development yields a representation of the number of (weighted) solutions within an appropriate domain that is very similar to that provided by the classical machinery, thus giving extra justification to the title Heath-Brown gives it. Yet the new representation has the advantage over the old one in that it no longer contains the sums $$Q_r ( {{{\bf{{m}}}}}, k)$$ for non-zero $$r$$ so that a Kloosterman refinement is latent within it. Thus, at the cost of an initial analysis more demanding than that previously needed, the new method is more favourable in two respects. First, the work of Heath-Brown and the author on denary and nonary cubic forms (Ha, Ib, Ic) could follow on from the first stages exactly as before, save that the sums $$Q_r ( {{{\bf{{m}}}}}, k)$$ for $$r \neq 0$$ could be ignored; however, nothing better would be achieved through this fact alone, and this later simplification could be regarded as being counter-balanced by the complexity of the initial development. Secondly, the absence of $$Q_r ( {{{\bf{{m}}}}}, k)$$ for $$r \neq 0$$ in the representation leads to the possibility of a double Kloosterman refinement along the lines of our Ia, the procedure of which we now adopt for octonary forms. Subject to the appropriate form of Hypothesis HW to be stated later, we shall show that a non-singular octonary form $$f( {{{\bf{{x}}}}})$$ with rational integral coefficients has a non-trivial integral zero provided that it have a non-trivial zero in every $$p$$-adic field $${\Bbb {Q}}_p$$. Because the structure of the proof becomes clear in the sequel, it is only appropriate to comment on two aspects of it now. One is that the Hypothesis HW is only used to improve estimates involving $$Q_0 ( {{{\bf{{m}}}}}, k)$$ for square-free values of $$k$$. With regard to higher powers of $$p$$ dividing a modulus $$k$$, there appear sums of the type $$Q_0 ( {{{\bf{{m}}}}}, p^{\alpha })$$ that require estimates much more accurate than those used in the previous papers Ha, Ib, Ic. In fact, granted the truth of Hypothesis HW, it is in the estimation of these sums that the greatest difficulty of this paper lies. This work has connections with Heath-Brown's [6], in which, among other things, he elaborates on our Ia (see also our [13]). But we do not refer to it further owing to exigencies of space and owing to the fact there is no mention there of the sums $$Q_0 ( {{{\bf{{m}}}}}, p^{\alpha })$$ for general forms $$f( {{{\bf{{x}}}}})$$. We make copious references to Ia, Ib, and Ic, on which many of our auxiliary results are based. So far as Heath-Brown's method concerns us, the only part in his Hb we use are his § §3–5. Anything else associated with his method is derived either directly or from those paragraphs in order to present the reader with a simplification of what he does. Note: In the text, papers [4] and [5] by Heath-Brown are indicated by Ha and Hb, respectively; papers [8], [9], [10], and [12] by Ia, Ib, Ic, and Id, respectively. 2. Notation The order $$n$$ of the form $$f( {{{\bf{{x}}}}})$$ in (1) will eventually be $$8$$, although most of the calculations will be valid for $$n \geq 6$$. The letter $$X$$ denotes a positive real variable that is to be regarded as tending to infinity, all inequalities that are valid for sufficiently large $$X$$ being assumed to hold. Positive constants depending on the given form $$f( {{{\bf{{x}}}}})$$ are denoted by $$A$$, $$A_1$$, $$A_2, \dotsc$$, etc. a number denoted by $$A$$ not necessarily being the same on all occasions; $$\epsilon$$ is an arbitrarily small positive number whose value may change according to well-known conventions. The letter $$p$$ denotes a (positive) prime number, $${\Bbb {F}}_p$$ being the prime field containing $$p$$ elements; we do not distinguish notationally between an integer $$m$$ and the element in $${\Bbb {F}}_p$$ to which it corresponds. Ordered $$n$$-tuples are indicated by bold type, their components being denoted by the same letter in italic font with subscripts; if $$ {{{\bf{{a}}}}} = ( {{a}_{1}, \dotsc , {a}_{{n}}})$$, then $$||{ {{{\bf{{a}}}}} }|| = \max |{a_i}|$$; $$ {{{\bf{{a}}}}} {{{\bf{{b}}}}}$$ is the scalar product $$a_1 b_1 + \dotsb +a_n b_n$$. Usually, these $$n$$-tuples are to be regarded as row vectors but on occasion they are column vectors when they participate in matrix algebra, as for example in the congruence following (45). Also the notation $$u <{{{\bf{{a}}}}} \leq v$$ signifies the condition that $$u <a_i \leq v$$ for $$1 \leq i \leq n$$; $$\boldsymbol {{\upsilon }}$$ is the unit vector $$(1, \dotsc , 1)$$; the zero vector is denoted by $$0$$ instead of $$\boldsymbol {0}$$. We use   \[\int g( {{{\bf{{w}}}}}) \, {d} {{{\bf{{w}}}}}\] to indicate   \[\int g( {{{\bf{{w}}}}}) \, {d} w_1 \dotsb {d} w_n,\] and use the upper and lower limits $$ {{{\bf{{a}}}}} +r {\boldsymbol {{\upsilon }}}$$, $$ {{{\bf{{a}}}}} - r {\boldsymbol {{\upsilon }}}$$ in the former expression when the domain of integration is $$||{ {{{\bf{{w}}}}}|| - {{{\bf{{a}}}}} } \leq r$$. The highest common factor and lowest common denominator of $$a$$ and $$b$$ are $$ {({a}, {b})}$$ and $$[a, b]$$; $$l {{\vert }} m$$ means $$l$$ divides $$m$$, while $$l^{\alpha } {{\parallel } } m$$ means that $$l^{\alpha }$$ is the highest power of $$l$$ that divides $$m$$; $$\omega (m)$$ is the number of distinct prime factors of $$m$$ and $$d(m)$$ is the number of positive divisors of $$m.$$ 3. Geometrical prolegomena and the basic dichotomy As already described, the cubic form   \[f( {{{\bf{{x}}}}}) = f ( {{x}_{1}, \dotsc , {x}_{{n}}}) = \sum _{{1 \leq i,j,k \leq n}} c_{i,j,k} x_i x_j x_k \quad {\mathrm {(}} {c_{i,j,k}}{\mathrm {\ symmetric)}}\] has rational integral coefficients (It is immaterial whether the $$c_{i,j,k}$$ or their multiples affected by the apposite binomial coefficients be assumed to be integers. We take the former interpretation here.) and belongs to a non-singular hypersurface $$ {{\cal {{{V}}}}}$$ of projective dimension $$n-2$$, where $$n \geq 6$$ although ultimately $$n$$ will be $$8$$ during the later stages of our demonstration. Among the concomitants of $$f$$, there are its non-vanishing discriminant $$D$$, its Hessian   \[H ( {{{\bf{{x}}}}}) = |{ {{{\bf{{M}}}}} ( {{{\bf{{x}}}}})}|\] that is the determinant of the symmetric matrix   \[ {{{\bf{{M}}}}} ( {{{\bf{{x}}}}}) = \left (\frac {\partial {^{2}}f}{\partial {x_j}\partial {x_k}}\right ) = 6 \left [{{{\sum _{1 \leq i \leq n}} c_{i,j,k}\ x_i}}\right ] \quad (1 \leq j, k \leq n),\] (2) and the contravariant $$F( {{{\bf{{u}}}}})$$ that is the form belonging to the dual hypersurface $$ {{\cal {{{V}}}}_{{\mathrm {dual}}}}$$ of $$ {{\cal {{{V}}}}}$$. As described in Ib, §8, the contravariant arises from the tangential equation of $$ {{\cal {{{V}}}}}$$ that is the result of eliminating $$\lambda$$, $$ {{x}_{1}, \dotsc , {x}_{{n}}}$$ from the $$n+1$$ homogeneous equations inherent in the system   \[{\mathrm {grad}} f( {{{\bf{{x}}}}}) = \lambda ^2 {{{\bf{{u}}}}}, {{{\bf{{u}}}}} {{{\bf{{x}}}}} = 0 \quad ( {{{\bf{{u}}}}} \neq 0)\] (3) to form a resultant that is the square of a form $$F( {{{\bf{{u}}}}})$$. Alternatively, we see that the condition that $$ {{{\bf{{u}}}}}$$ be on $$ {{\cal {{{V}}}}_{{\mathrm {dual}}}}$$ is that the section $$ {{\cal {{{V}}}}} ( {{{\bf{{u}}}}}) = {{\cal {{{V}}}}} \cap ( {{{\bf{{u}}}}} {{{\bf{{x}}}}} = 0)$$ of $$ {{\cal {{{V}}}}}$$ cut by the hyperplane $$ {{{\bf{{u}}}}} {{{\bf{{x}}}}} = 0$$ be singular, in which case the singular locus of $$ {{\cal {{{V}}}}} ( {{{\bf{{u}}}}})$$ has dimension zero by a principle of Zack's (for a simple proof, see Lemma 39 of Ic) and therefore consists of not more than $$A$$ points. The Hessian matrix $$ {{{\bf{{M}}}}} ( {{{\bf{{x}}}}})$$ will play a significant rôle in several aspects of our treatment. In particular, among its properties we later use, it enjoys the well-known feature, confirmed anew in §3 of Ib, that $$H( {{{\bf{{x}}}}})$$ does not vanish identically on $$ {{\cal {{{V}}}}}$$, there thus being a real point on $$ {{\cal {{{V}}}}}$$ having a non-zero Hessian. This formed a basis of our self-contained demonstration of a classical relationship between $$ {{\cal {{{V}}}}}$$ and its dual that was stated in Lemma 2 of Ib and that we repeat here as follows: Lemma 1 Let$$ {{{\bf{{u}}}}} = {\mathrm {grad}} f( {{{\bf{{x}}}}})$$where$$f( {{{\bf{{x}}}}}) = 0$$. Then, for some$$\lambda \neq 0,$$  \[ {{{\bf{{x}}}}} = \lambda \ {\mathrm {grad}} \ F( {{{\bf{{u}}}}})\]provided that$${\mathrm {grad}} F( {{{\bf{{u}}}}}) \neq 0,$$namely, that$$ {{{\bf{{u}}}}}$$be not a singular point on$$ {{\cal {{{V}}}}_{{\mathrm {dual}}}}.$$ On introducing the indeterminate equation   \[F( {{{\bf{{m}}}}}) = 0\] (4) that may be thought of as the dual of $$f( {{{\bf{{l}}}}}) = 0$$, we see from the lemma that any non-singular solution $$ {{{\bf{{m}}}}}$$ of (4) produces a non-trivial integral zero of $$f( {{{\bf{{x}}}}})$$. Hence, in pursuing our aim of showing that (1) is soluble subject to the appropriate $$p$$-adic condition, we may now assume that   \[{\mathrm {`all\ solutions\ of\ {(4)}{}\ are\ singular'}},\] (5) which restriction makes possible the success of the analytical method that follows but which rules out our obtaining a weak approximation result here. It is indeed not without the bounds of possibility that the methods used in our paper Id could be used to remove this shortcoming. But we prefer not to explore this matter on the present occasion because the likely complexity of what might be involved would add an unacceptable extra length to this paper. So far our remarks have been made on the tacit understanding that the prime field underlying them has been the rational field $${\Bbb {Q}}$$. But many of them are yet appropriate when $${\Bbb {Q}}$$ is replaced by the finite field $${\Bbb {F}}_p$$ when $$p >p_0$$ and especially when we consider the reduction $$f( {{{\bf{{x}}}}}), {\hbox{mod}\, {p}}$$, of $$f( {{{\bf{{x}}}}})$$ that is the defining form for the reduction $$ {{\cal {{{V}}}}}_p$$ of $$ {{\cal {{{V}}}}}$$. First, since $$D \neq 0$$, for $$p >p_0$$ we have that $$D \not\equiv 0, {\hbox{mod}\, {p}}$$, and that therefore $$ {{\cal {{{V}}}}}_p$$ is non-singular; also by interpreting (3) to the modulus $$p$$ or as a set of equations over $${\Bbb {F}}_p$$ when $$ {{{\bf{{u}}}}}$$ is a vector with components in $${\Bbb {F}}_p$$, we see that the counterpart of the original $$F( {{{\bf{{u}}}}})$$ reached through the elimination process can be written as $$F( {{{\bf{{u}}}}}), {\hbox{mod}\, {p}}$$, or $$F( {{{\bf{{u}}}}})$$ according to our notational conventions, or, in other words, the dual of $$ {{\cal {{{V}}}}}_p$$ is $$( {{\cal {{{V}}}}_{{\mathrm {dual}}}})_p$$. This implies then, in particular, that $$ {{\cal {{{V}}}}}_p ( {{{\bf{{m}}}}}) = {{\cal {{{V}}}}}_p \cap ( {{{\bf{{m}}}}} {{{\bf{{x}}}}} = 0)$$ is singular if and only if $$F( {{{\bf{{m}}}}}) \equiv 0, {\hbox{mod}\, {p}}$$, the previous comments about the number of singularities in a singular section being still applicable. Finally, just as the known absolute irreducibility of $$F( {{{\bf{{u}}}}})$$ is established, so can that of it over $${\Bbb {F}}_p$$ for $$p >p_0$$ be confirmed, although alternatively the latter can be deduced from the former by appealing to Noether's criterion for the absolute irreducibility of forms. Consequently, the singular locus of the dual of $$ {{\cal {{{V}}}}}_p$$ has dimension (projective) not more than $$n-3$$. (In fact it is known that the dimension is exactly $$n-3$$. See the beginning paragraph of §  11.) Another property of the matrix $$ {{{\bf{{M}}}}} ( {{{\bf{{x}}}}})$$ on which we shall draw is stated in the following lemma: Lemma 2 Let$$ {{\cal {{{V}}}}}_{r,p}^{ {\dagger } }$$for$$p >p_0$$be the affine variety defined over$${\Bbb {F}}_p$$consisting of the points$$ {{{\bf{{x}}}}}$$for which$$f( {{{\bf{{x}}}}}) = 0$$and the determinantal rank of$$ {{{\bf{{M}}}}} ( {{{\bf{{x}}}}})$$does not exceed$$r$$. Then  \[\dim \, {{\cal {{{V}}}}}_{r,p}^{ {\dagger } } \leq r-1\] (6)for$$r=1,$$$$n-1,$$and$$n,$$while  \[\dim \, {{\cal {{{V}}}}}_{r,p}^{ {\dagger } } \leq r\] (7)otherwise. That (6) for $$r=n-1$$, $$n$$ and (7) for $$1 \leq r \leq n-2$$ are true is clear from the proof of Lemma 36 in Ic, since the method there is unaffected from the change of $$ {{\cal {{{V}}}}}$$ as a hypersurface with isolated double points to one that is non-singular. We therefore only need to show that $$\dim {{\cal {{{V}}}}}_{r,p}^{ {\dagger } } =1$$ is impossible when $$r=1$$. If the opposite were possible, then there would be a point $$ {{{\bf{{a}}}}}$$ (corresponding to a one-dimensional ray) for which $$f( {{{\bf{{a}}}}}) = 0$$ and $${\mathrm {rank}} {{{\bf{{M}}}}} ( {{{\bf{{a}}}}}) \leq 1$$, wherefore there would be a unimodular transformation of coordinates that takes in turn $$ {{{\bf{{a}}}}}$$ into $$(1,0, \ldots , 0)$$, $$f( {{{\bf{{x}}}}})$$ into a non-singular form   \[f^{(1)} ( {{{\bf{{y}}}}}) = \sum _{{1 \leq i,j,k \leq n}} c_{i,j,k}^{(1)} y_i y_j y_k,\] and the covariant matrix $$ {{{\bf{{M}}}}} ( {{{\bf{{x}}}}})$$ into the Hessian matrix $$ {{{\bf{{M}}}}} ^{(1)} ( {{{\bf{{y}}}}})$$ of $$f^{(1)} ( {{{\bf{{y}}}}})$$. Hence, since $$f^{(1)} (1,0, \ldots , 0) = 0$$, we would infer that $$c_{1,1,1}^{(1)} = 0$$ and then from (2) that   \[\frac {1}{6} {{{\bf{{M}}}}} ^{(1)} (1,0, \ldots , 0) = [{c_{1,j,k}^{(1)}}] \quad (1 \leq j,k \leq n)\] is a symmetric matrix of rank not exceeding $$1$$ whose leading element is zero. Forming the symmetric minor of order $$2$$ from this matrix by taking the constituents $$c_{1,1,1}^{(1)}$$, $$c_{1,1,k}^{(1)}$$, $$c_{1,k,1}^{(1)}$$, and $$c_{1,k,k}^{(1)}$$, we deduce that   \[c_{1,1,1}^{(1)} c_{1,k,k}^{(1)} - (c_{1,1,k}^{(1)})^{2} = - (c_{1,1,k}^{(1)})^{2} = 0\] and hence that all the coefficients in $$f^{(1)} ( {{{\bf{{y}}}}})$$ having at least two subscripts equal to $$1$$ are zero. Consequently, we reach the impossible conclusion that $$f^{(1)} ( {{{\bf{{y}}}}})$$ would be a singular form of the shape   \[y_1 Q(y_2, \ldots , y_n) +C(y_2, \ldots , y_n),\] the proof of the lemma being therefore complete. We should add the comment that, if, as seems likely, (6) be actually true for all values of $$r$$ between $$1$$ and $$n$$, then certain aspects of the coming treatment could be significantly simplified. But the proof of this supposition, if true, would seem to be difficult and the most we have achieved in this direction is to establish it for a generic cubic form, that is, one whose coefficients do not satisfy any of a finite number of polynomial equations. This can be done, for example, by confirming that the supposition is vindicated when $$f( {{{\bf{{x}}}}})$$ is a diagonal form. 4. The initial analysis: the application of the Heath-Brown form of the circle method To bring in the analytical method on the assumption of (5) we let   \[\Gamma ( {{{\bf{{t}}}}}) = \prod _{{1 \leq i \leq n}} \gamma (t_i),\] where   \[\gamma (t) = \left\{ \begin {array}{ll}e^{-2 / (1-t^2)} & {\mathrm {if\ }} |{t}| <1, \\ 0 & {\mathrm {if\ }} |{t}| \geq 1, \end {array}\right .\] and then use the discussion in the previous section to choose a real fixed point $$ {{{\bf{{a}}}}}$$ on $$ {{\cal {{{V}}}}} ^{ {\dagger } }$$ that is a sufficiently large scalar multiple $$\lambda {{{\bf{{a}}}}} ^{\prime }$$ of a given real point $$ {{{\bf{{a}}}}} ^{\prime }$$ on $$ {{\cal {{{V}}}}} ^{ {\dagger } }$$ such that $$H( {{{\bf{{a}}}}} ^{\prime }) \neq 0$$. Chosen for their smoothing effects, these entities appear in the sum   \[\Upsilon (X) = \sum _{{f ( {{{\bf{{l}}}}}) = 0}} \Gamma \left({\frac { {{{\bf{{l}}}}} }{X} - {{{\bf{{a}}}}} }\right)\] that counts with a certain weight the number of solutions of the indeterminate equation in (1) in a large hyperparallelepiped with sides of length $$2X$$. This is an example of the sums $$N^{(0)} (F, w)$$ that are the subject of Theorem 2 of Hb and is therefore determined by   \[\Upsilon (X) = \frac {c_N}{N^2} \sum _{ {{{\bf{{m}}}}} } \sum _{k=1}^{\infty } \frac {Q ( {{{\bf{{m}}}}}, k)}{k^n} I_{k}^{(0)} ( {{{\bf{{m}}}}})\] (8) for any value of $$N$$ exceeding $$1,$$ where $$Q ( {{{\bf{{m}}}}}, k)$$ is the important trigonometrical sum   \[\sum _{\substack {0 <h \leq k \\ {({h}, {k})} = 1}} {\sum _{0 <{{{\bf{{l}}}}} \leq k}} e^{2 \pi i (h f( {{{\bf{{l}}}}}) - {{{\bf{{m}}}}} {{{\bf{{l}}}}})/k},\] (9)  \[I_{k}^{(0)} ( {{{\bf{{m}}}}}) = \int _{( {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}}) X}^{( {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}}) X} \Gamma \left( {\frac { {{{\bf{{t}}}}} ^{\prime }}{X} - {{{\bf{{a}}}}} }\right) h \left( {\frac {k}{N}, \frac {f ( {{{\bf{{t}}}}} ^{\prime })}{N^2}}\right) e^{-2 \pi i {{{\bf{{m}}}}} {{{\bf{{t}}}}} ^{\prime } / k} {d} {{{\bf{{t}}}}} ^{\prime },\] and the entities $$h(x,y)$$ and   \[c_N = 1 + O \left({\frac {1}{N^{A}}}\right)\] (10) appear explicitly in §3 of Hb. Having aligned our notation to parallel that of Ia and Ib where also indeterminate equations of the type $$f( {{{\bf{{l}}}}}) = 0$$ were considered, we are thus prepared to avail ourselves easily of the results of our earlier papers but note that our previous sum $$Q_{0} ( {{{\bf{{m}}}}}, k)$$ is now written $$Q ( {{{\bf{{m}}}}}, k)$$ because the use of Heath-Brown's method eliminates the presence in the analysis of the sums $$Q_{r} ( {{{\bf{{m}}}}}, k)$$ when $$r \neq 0$$. Let us develop (8) by further setting   \[N = X^{{3}/{2}}\] (11) and transforming $$I_{k}^{(0)} ( {{{\bf{{m}}}}})$$ into   \[X^n \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( { {{{\bf{{t}}}}} - {{{\bf{{a}}}}} }) h \left({\frac {k}{N}, f ( {{{\bf{{t}}}}})}\right) \,e^{-2 \pi i X {{{\bf{{m}}}}} {{{\bf{{t}}}}} / k} \, {d} {{{\bf{{t}}}}} = X^n I_{k} ( {{{\bf{{m}}}}}) \quad {\mathrm {say}},\] (12) the integrand in the last integral being zero for   \[k >A_1 N\] (13) by Lemma 4 of Hb because $$f ( {{{\bf{{t}}}}})$$ is bounded over the region of integration. Therefore,   \[\begin {array}{rl} \Upsilon (X) & \geq \frac {1}{2} X^{n-3} \sum _{ {{{\bf{{m}}}}} } \sum _{k \leq A_1 N} \frac {Q ( {{{\bf{{m}}}}}, k)}{k^n} I_{k} ( {{{\bf{{m}}}}}) \\ & = \frac {1}{2} X^{n-3} \left ( \sum _{k \leq A_1 N} \frac {Q (0, k)}{k^n} I_{k} (0) + \sum _{F( {{{\bf{{m}}}}}) \neq 0} \sum _{k \leq A_1 N} \frac {Q ( {{{\bf{{m}}}}}, k)}{k^n} I_{k} ( {{{\bf{{m}}}}}) \right .\\ & \quad \left . + \sum _{\substack { {{{\bf{{m}}}}} \neq 0 \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0}} \sum _{k \leq A_1 N} \frac {Q ( {{{\bf{{m}}}}}, k)}{k^n} I_{k} ( {{{\bf{{m}}}}}) \right )\\ & = \frac {1}{2} X^{n-3} \{\Upsilon _1 (X) + \Upsilon _2 (X) + \Upsilon _3 (X)\} \quad {\mathrm {say}}, \end {array}\] (14) in virtue of our assumption (5). The most important term in this inequality is $$\Upsilon _1 (X)$$ in the sense that it dominates the right-hand side, although from our point of view the other two terms are the most significant because of the difficulties encountered in their estimation. We therefore focus on $$\Upsilon _2 (X)$$ and $$\Upsilon _3 (X)$$ for the better part of the memoir, leaving the comparatively simple treatment of $$\Upsilon _1 (X)$$ to the end. Moreover, in treating the former sums we shall direct our attention for the most part on their segments $$\Psi _i (Y) = \Psi _i (Y,X)$$ that correspond to those values of $$k$$ in the summations for which $$\frac {1}{2} Y <k \leq Y$$, where of course   \[Y \leq A_1 N.\] (15) by (13). Before going on, we should observe that (14) is similar but simpler in appearance to the expression for $$\Upsilon (X)$$ in Ib and Ic when the order of the Farey dissection used therein is close to $$N$$. Yet, as a penalty for this apparent simplicity, there now appears the integral $$I_{k} ( {{{\bf{{m}}}}})$$ that is not closely analogous to anything present in the former papers and that is a major source of difficulty because of the presence of Heath-Brown's function $$h(x,y)$$. This integral must therefore be investigated in the next section before the method is further unrolled. 5. The integrals $$I_k( {{{\bf{{m}}}}})$$ and their presence in certain sums Since Heath-Brown's treatment of these integrals is difficult in places, we prefer an alternative approach that still depends on his analysis of the function $$h(x,y)$$ but also uses estimates for the sum $$H(u, {{{\bf{{v}}}}})$$ in our paper Ib. First there is the following lemma: Lemma 3 We have  \[I_{k} ( {{{\bf{{m}}}}}) = O(1).\]Also, if  \[ {{{\bf{{v}}}}} = {{{\bf{{m}}}}} X / k \quad {\mathrm {and}}\quad ||{ {{{\bf{{v}}}}} }|| \leq A\] (16)for any positive constant$$A$$, then  \[k \frac{\partial I_{k} ( {{{\bf{{m}}}}})}{\partial k} = O(1).\] Letting   \[r = k / N \leq A,\] (17) as we may by (13), we use the bounds   \[h(r,f ( {{{\bf{{t}}}}})) = O \left\{ \frac {1}{r} \left({r^2 + {{\min}^{{2}}} \left({1, \frac {r}{ |{f( {{{\bf{{t}}}}})}| }}\right) } \right)\right\}\] (18) and   \[\frac{\partial h(r, f( {{{\bf{{t}}}}}))}{\partial r} = O \left\{ \frac {1}{r^2} \left({r^2 + {{\min}^{{2}}} \left({1, \frac {r}{ |{f( {{{\bf{{t}}}}})}| }}\right)} \right)\right\}\] (19) of Lemma 5 of Hb. At the point $$ {{{\bf{{a}}}}} ^{\prime }$$ chosen at the beginning of §4 we may assume that for some index $$j$$ we have $$ {\partial {^{{}}}{f ( {{{\bf{{a}}}}} ^{\prime })} / \partial {{a_j}^{{}}}} \neq 0$$ by the non-singularity of $$ {{\cal {{{V}}}}}$$, whence by an appropriate choice of $$\lambda$$ we have $$|{ {\partial {^{{}}}{f( {{{\bf{{t}}}}})} / \partial {{t_j}^{{}}}} }| >1$$ throughout the region of integration in $$I_{k} ( {{{\bf{{m}}}}})$$, the measure of points within for which $$|{f( {{{\bf{{t}}}}})}| \leq R$$ being therefore $$O(R)$$. Hence,   \[|{I_{k} ( {{{\bf{{m}}}}})}| \leq \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } |{h(r,f( {{{\bf{{t}}}}}))}| {d} {{{\bf{{t}}}}} = O(1) +O(1) + {O \left({{r \int _{{\substack { |{f( {{{\bf{{t}}}}})}| \geq r \\ ||{ {{{\bf{{t}}}}} - {{{\bf{{a}}}}} }|| \leq 1}}} \frac { {d} {{{\bf{{t}}}}} }{ |{f( {{{\bf{{t}}}}})}| ^2}}}\right)},\] (20) in which equation, by setting   \[g(u) = \int _{{\substack { |{f( {{{\bf{{t}}}}})}| \leq u \\ ||{ {{{\bf{{t}}}}} - {{{\bf{{a}}}}} }|| \leq 1}}} {d} {{{\bf{{t}}}}} = O({u}),\] we see that the integral on the right is equal to   \[\int _{r}^{\infty } \frac { {d} g(u)}{u^2} = {}^{\infty }_{r}{ \left [{{\frac {g(u)}{u^2}}}\right ] } +2 \int _{r}^{\infty } \frac {g(u) \, {d} u}{u^3} = {O \left({{\frac {1}{r}}}\right)} + {O \left({{\frac {1}{r}}}\right)} = {O \left({{\frac {1}{r}}}\right)}.\] The first part of the lemma then follows. Also   \[\begin {array}{rl} k \frac{\partial I_{k} ( {{{\bf{{m}}}}})}{\partial k} & = \frac {k}{N} \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( { {{{\bf{{t}}}}} - {{{\bf{{a}}}}} }) \frac {\partial h(r,f( {{{\bf{{t}}}}}))}{\partial r} e^{-2 \pi i X {{{\bf{{m}}}}} {{{\bf{{t}}}}} / k} \, {d} {{{\bf{{t}}}}} \\ & \quad + \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \left({ \frac {2 \pi i X {{{\bf{{m}}}}} {{{\bf{{t}}}}} }{k}}\right) \Gamma ( { {{{\bf{{t}}}}} - {{{\bf{{a}}}}} }) h(r,f( {{{\bf{{t}}}}})) e^{-2 \pi i X {{{\bf{{m}}}}} {{{\bf{{t}}}}} / k} \, {d} {{{\bf{{t}}}}} \\ & = r I_{k}^{ * } ( {{{\bf{{m}}}}}) +I_{k}^{ * * } ( {{{\bf{{m}}}}}) \quad {\mathrm {say}}, \end {array}\] (21) which equation may be expressed as   \[ {O \left({{r \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \left| {\frac {\partial h(r,f( {{{\bf{{t}}}}}))}{\partial r}}\right| {d} {{{\bf{{t}}}}} }}\right)} + {O \left({{\frac {X ||{ {{{\bf{{m}}}}}||}} {k} \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } |{h(r,f( {{{\bf{{t}}}}}))}| {d} {{{\bf{{t}}}}} }}\right)}.\] Here,   \[I_{k}^{ * * } ( {{{\bf{{m}}}}}) = {O \left({{\frac {X ||{ {{{\bf{{m}}}}} } ||}{k}}}\right)} = O(1)\] by the estimations connected with (20) above, while   \[I_{k}^{ * } ( {{{\bf{{m}}}}}) = {O \left({{\frac {r}{r}}}\right)} = O(1)\] by using (19) instead of (18) within the method used to establish the first part of the lemma; the proof of the second part of the lemma is therefore complete. The next properties we require of $$I_{k} ( {{{\bf{{m}}}}})$$ lie deeper than Lemma 3 and are established with the aid of the following estimates for certain integrals containing exponential sums that were partially obtained in Ib. Lemma 4 Let  \[H (u, {{{\bf{{v}}}}}_{1}) = \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) \,e^{2 \pi i (u f( {{{\bf{{t}}}}}) + {{{\bf{{v}}}}}_{1} {{{\bf{{t}}}}})} {d} {{{\bf{{t}}}}}\]and  \[H_1 (u, {{{\bf{{v}}}}}_{1}) = \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \left({{\sum _{1 \leq i \leq n}} t_i \frac{\partial \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}})}{\partial t_i}}\right) e^{2 \pi i (u f( {{{\bf{{t}}}}}) + {{{\bf{{v}}}}}_{1} {{{\bf{{t}}}}})} {d} {{{\bf{{t}}}}}.\]Then  \[H (u, {{{\bf{{v}}}}}_{1}), H_1 (u, {{{\bf{{v}}}}}_{1}) = \left\{ \begin {array}{ll}O({ |{u}| ^{- ({1}/{2})n} \log ^{n} 2 |{u}| }) & {\mathrm {if\ }} |{u}| \geq 1, \\ O({e^{- A_2 ||{ {{{\bf{{v}}}}}_{1}}||^{{1}/{2}}}}) & {\mathrm {if\ }} ||{ {{{\bf{{v}}}}}_{1}}|| >A_3 |{u}|, \\ O(1) & {\mathrm {always}}, \end {array}\right .\]where$$A_3$$can be taken to exceed$$1$$. The bounds for $$H (u, {{{\bf{{v}}}}}_{1})$$ are given in Lemma 7 of Ib, while those for $$H_1 (u, {{{\bf{{v}}}}}_{1})$$ are obtained in a similar manner. We shall use Fourier's integral theorem to express the functions   \[\left. h(r, f( {{{\bf{{t}}}}})), \frac{\partial h(r, f( {{{\bf{{t}}}}}))}{\partial r},y \frac{\partial h(r,y)}{\partial y}\right|_{y = f( {{{\bf{{t}}}}})}\] in terms of Fourier transforms for points $$ {{{\bf{{t}}}}}$$ within the region $$||{ {{{\bf{{t}}}}} - {{{\bf{{a}}}}} }|| \leq 1$$. But, since the nature of $$h(r,y)$$ and its partial derivatives is such that their transforms are divergent when these are taken with respect to all $$y$$, we replace $$h(r,y)$$ by an infinitely differentiable function $$h_1 (r,y)$$ that equals $$h(r,y)$$ for values of $$y$$ that include those taken by $$f( {{{\bf{{t}}}}})$$ in the above-mentioned region of $$ {{{\bf{{t}}}}}$$ but that vanishes for all sufficiently large $$y$$. To be precise, since $$|{f( {{{\bf{{t}}}}})}| <A$$ within this region, we construct an infinitely differentiable function $$g(y)$$ having the properties   \[g(y) = 1 {\mathrm {\ for\ }} |{y}| \leq A, \quad g(y) \downarrow {\mathrm {\ for\ }} A <|{y}| \leq A +1, \quad g(y) = 0 {\mathrm {\ for\ }} |{y}| \geq A +1\] and then set   \[h_1 (r,y) = g(y) h(r,y),\] which function possess the features we need. Since $$r = O(1)$$, Leibnitz's theorem and Lemma 5 of Hb with $$N=3$$ and 2 easily yield the respective bounds   \[\begin {array}{rl} {\partial^{M} \ h_1 (r,y)}{\partial y^{M}} & = O({r^2}) +O \left\{ \frac {1}{r^{M+1}} {{\min}^{{3}}} \ {\left(1, \frac {r}{ |{y}| }\right)}\right\} \\ & = O({r}) +O \left\{\frac {1}{r^{M+1}} {{\min}^{{2}}} \ {\left(1, \frac {r}{ |{y}| }\right)}\right\} \end {array}\] (22) and   \[\frac{\partial^{M+1}h_1 (r,y)}{\partial r \partial y^{M}} = O(1) +O \left\{\frac {1}{r^{M+2}} {{\min}^{{2}}} \ {\left(1, \frac {r}{ |{y}| }\right)}\right\}, \] (23) from the former of which it is inferred that   \[\begin {array}{rl} \frac{\partial^{M}}{\partial y^{M}} \left(y \frac{\partial h_1 (r,y)}{\partial y}\right) & = O({r^2 |{y}| }) +O \left\{\frac{ |{y}| }{r^{M+2}} {{\min}^{{3}}} \ \left(1, \frac {r}{ |{y}| }\right)\right\} \\ &\quad +O({r}) +O \left\{ \frac {1}{r^{M+1}} {{\min}^{{2}}} \ \left(1, \frac {r}{ |{y}| }\right)\right\} \\ & = O({r}) +O \left\{ \frac {1}{r^{M+1}} {{\min}^{{2}}} \ \left(1, \frac {r}{ |{y}| }\right)\right\} \end {array}\] (24) since it may be assumed that $$y = O(1)$$. If $$p_1 (u)$$ be the Fourier transform of $$h_1 (r,y)$$, then by partial integration   \[p_1 (u) = \int _{-A-1}^{A+1} h_1 (r,y) e^{2 \pi i u y} {d} y = \frac {1}{(2 \pi i u)^{M}} \int _{-A-1}^{A+1} \frac {\partial {^{M}}h_1 (r,y)}{\partial {y^{M}}} e^{2 \pi i u y} {d} y\] for $$u \neq 0$$ and any large integer $$M$$, since all partial derivatives of $$h_1 (r,y)$$ vanish for $$|{y}| >A +1$$. Hence, by (22),   \[p_1 (u) = O \left\{\frac {1}{u^{M}} \int _{0}^{A+1} \left(r + \frac {1}{r^{M+1}} {{\min}^{{2}}} \ \left(1, \frac {r}{y}\right)\right) \ dy\right\} = O \left(\frac {1}{(r u)^{M}}\right)\] (25) for $$u \neq 0$$, while also   \[p_1 (u) = O \left(\int _{-A-1}^{A+1} |{h_1 (r,y)}| {d} y\right) = O \left\{ \int _{0}^{A+1} \left(r + \frac {1}{r} {{\min}^{{2}}} \ \left(1, \frac {r}{y}\right)\right)dy\right\} = O(1)\] (26) always. In like manner the respective Fourier transforms $$p_2 (u)$$ and $$p_3 (u)$$ of $$r {\partial {^{{}}}{h_1 (r,y)} / \partial {{r}^{{}}}}$$ and $$y {\partial {^{{}}}{h_1 (r,y)} / \partial {{y}^{{}}}}$$ are subject to the bounds in the right sides of (25) and (26) because of (23) and (24). We are ready to supplement Lemma 3 by more delicate estimates, to which end it is temporarily convenient to suppose that   \[||{ {{{\bf{{v}}}}} }|| >A_3\] (27) in contrast with (16) where $$A = A_3$$ in the statement of Lemma 4. Then, since   \[h_1 (r,y) = \int _{-\infty }^{\infty } p_1 (u) \,e^{- 2 \pi i u y} \, {d} u\] and $$h (r, f( {{{\bf{{t}}}}})) = h_1 (r, f( {{{\bf{{t}}}}}))$$ within the region of integration in $$I_{k} ( {{{\bf{{m}}}}})$$, we have   \[\begin {array}{rl} I_{k} ( {{{\bf{{m}}}}}) & = \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) e^{- 2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } \int _{-\infty }^{\infty } p_1 (u) e^{- 2 \pi i u f( {{{\bf{{t}}}}})} \, {d} u \, {d} {{{\bf{{t}}}}} \\ & = \int _{-\infty }^{\infty } p_1 (u) \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) e^{- 2 \pi i {\{ }u f( {{{\bf{{t}}}}}) + {{{\bf{{v}}}}} {{{\bf{{t}}}}} {\} }} \, {d} {{{\bf{{t}}}}} \, {d} u \\ & = \int _{-\infty }^{\infty } p_1 (u) H(-u, - {{{\bf{{v}}}}}) \, {d} u, \end {array}\] there being two cases to consider according as   \[\frac {1}{r} <\frac {||{ {{{\bf{{v}}}}} }|| }{A_3} \quad {\mathrm {or}}\quad \frac {1}{r} \geq \frac {||{ {{{\bf{{v}}}}} }|| }{A_3}.\] (28) In the former case, since $$A_3^{-1} ||{ {{{\bf{{v}}}}} }|| \geq 1$$ by (27), we have from (25), (26), and Lemma 4 that   \[\begin {array}{rl} I_k ( {{{\bf{{m}}}}}) & = {O \left({{\int _{0}^{A_3^{-1} ||{ {{{\bf{{v}}}}}} ||} e^{- A_2 ||{ {{{\bf{{v}}}}} }||^{{1}/{2}}} {d} v}}\right)} + {O \left({\frac {1}{r^{M}} \int _{A_{3}^{-1} ||{ {{{\bf{{v}}}}} }|| }^{\infty } \frac {\log ^n 2u}{u^{M + ({1}/{2}) n}} {d} u}\right)} \\ & = O(e^{- A ||{ {{{\bf{{v}}}}} }||^{{1}/{2}}}) + {O \left({{\frac {\log ^n 2 ||{ {{{\bf{{v}}}}} }|| }{||{ {{{\bf{{v}}}}} }||^{({1}/{2}) n - 1} \ (r ||{ {{{\bf{{v}}}}} }||)^{M}}}}\right)} \\ & = {O \left({{\frac {\log ^n 2 ||{ {{{\bf{{v}}}}}}||}{|| { {{{\bf{{v}}}}} }||^{({1}/{2}) n - 1} (r ||{ {{{\bf{{v}}}}} }||)^{M}}}}\right)} \end {array}\] (29) for any positive integer $$M$$. But in the latter case   \[\begin {array}{rl} I_k ( {{{\bf{{m}}}}}) & = {O \left({{\int _{0}^{A_3^{-1} ||{ {{{\bf{{v}}}}} } ||} e^{- A_2 ||{ {{{\bf{{v}}}}} }||^{{1}/{2}}} {d} u}}\right)} + {O \left({{\int _{A_3^{-1} ||{ {{{\bf{{v}}}}}}||}^{\infty } \frac {\log ^n 2u}{u^{({1}/{2}) n}} {d} u}}\right)} \\ & = {O \left({{\frac {\log ^n 2 ||{ {{{\bf{{v}}}}} } ||}{||{ {{{\bf{{v}}}}} }||^{({1}/{2}) n - 1}}}}\right)} \end {array}\] (30) by similar reasoning. Here by (17) and the definition of $$ {{{\bf{{v}}}}}$$ in (16) we note that (29) or (30) is apposite for $$||{ {{{\bf{{m}}}}} }|| >A_3 k / X$$ according as $$||{ {{{\bf{{m}}}}} }|| >A_3 N / X$$ or $$||{ {{{\bf{{m}}}}} }|| \leq A_3 N / X$$, although the division between the cases is obviously subject to some latitude. To bound $$k {\partial {^{{}}}{I_k ( {{{\bf{{m}}}}})} / \partial {{k,}^{{}}}}$$ we return to (21), in which the constituent   \[r I_k^{ * } ( {{{\bf{{m}}}}}) = \int _{- \infty }^{\infty } r p_2(u) H(-u, - {{{\bf{{v}}}}}) \, {d} u\] is governed by bounds of type (29) or (30) because the transforms $$p_1(u)$$ and $$r p_2(u)$$ exhibit similar behaviour. On the other hand, to obtain like bounds for the second piece $$I_k^{ * * } ( {{{\bf{{m}}}}})$$ is a matter of some subtlety and depends in part on an artifice discovered by Heath-Brown and used by him in a slightly different context in §8 of Hb. Temporarily writing $$\Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) h(r, f( {{{\bf{{t}}}}}))$$ as $$g( {{{\bf{{t}}}}})$$ in the integrand of $$I_k^{ * * } ( {{{\bf{{m}}}}})$$, we consider   \[{\mathrm {div}} g( {{{\bf{{t}}}}}) e^{- 2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } {{{\bf{{t}}}}}\] and find that it is   \[(-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}}) g( {{{\bf{{t}}}}}) e^{-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } +n g( {{{\bf{{t}}}}}) e^{-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } + ( {{{\bf{{t}}}}}. {\mathrm {grad}} g( {{{\bf{{t}}}}})) e^{-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} },\] where   \[\begin {array}{rl} {{{\bf{{t}}}}}. {\mathrm {grad}} g( {{{\bf{{t}}}}}) & = ( {{{\bf{{t}}}}}. {\mathrm {grad}} \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}})) h(r, f( {{{\bf{{t}}}}})) + \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) {{{\bf{{t}}}}}. {\mathrm {grad}} f( {{{\bf{{t}}}}}) \left.\frac{\partial h(r,y)}{\partial y}\right|_{y = f( {{{\bf{{t}}}}})} \\ & = ( {{{\bf{{t}}}}}. {\mathrm {grad}} \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}})) h(r, f( {{{\bf{{t}}}}})) +3 \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) y \left. \frac{\partial h(r,y)}{\partial y}\right|_{y = f( {{{\bf{{t}}}}})} \end {array}\] since $$f( {{{\bf{{t}}}}})$$ is a form of degree $$3$$. Therefore, by the divergence theorem, we deduce that   \[\begin {array}{rl} I_k^{ * * } ( {{{\bf{{m}}}}}) & = n \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) h(r, f( {{{\bf{{t}}}}})) \,e^{-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } \, {d} {{{\bf{{t}}}}} \\ & \quad +3 \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) \ y \left.\frac {\partial h(r,y)}{\partial y}\right|_{y = f( {{{\bf{{t}}}}})} \,e^{-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } \, {d} {{{\bf{{t}}}}} \\ & \quad + \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } {{{\bf{{t}}}}}. {\mathrm {grad}} \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) h(r, f( {{{\bf{{t}}}}})) \,e^{-2 \pi i {{{\bf{{v}}}}} {{{\bf{{t}}}}} } \, {d} {{{\bf{{t}}}}}, \end {array}\] the first term on the right-hand side of which is $$-n I_k ( {{{\bf{{m}}}}})$$ and therefore bounded through (29) or (30). The other two terms are limited through the same bounds by using the previous Fourier integral method in conjunction with the stated bounds for $$p_1(u)$$ and $$p_3(u)$$ and the appropriate parts of Lemma 4, whereupon by (21) we see that $$I_k ( {{{\bf{{m}}}}})$$ and $$k {\partial {^{{}}}{I_k ( {{{\bf{{m}}}}})} / \partial {{k}^{{}}}}$$ can be majorized in the same way when (27) applies. What we have obtained from Lemma 3 onwards is best summarized by the introduction of the function   \[J ( {{{\bf{{m}}}}}, Y) = \left\{ \begin {array}{ll}1 & {\mathrm {if\ }} ||{ {{{\bf{{m}}}}} }|| \leq Y / X, \\ \log ^n (2 X ||{ {{{\bf{{m}}}}} }|| / Y) (Y / X ||{ {{{\bf{{m}}}}} }||)^{({1}/{2}) n -1} & {\mathrm {if\ }} Y / X <||{ {{{\bf{{m}}}}} }|| \leq N / X, \\ \log ^n (2 X ||{ {{{\bf{{m}}}}} }|| / Y) (Y / X ||{ {{{\bf{{m}}}}}}||)^{({1}/{2}) n -1} (N / X ||{ {{{\bf{{m}}}}} }||)^{M} & {\mathrm {if\ }} ||{ {{{\bf{{m}}}}} }|| >N / X, \end {array}\right .\] where $$M$$ is any given large integer. Then, for $$\frac {1}{2} Y \leq k <Y$$ and $$Y \leq N = A_1 X^{{3}/{2}}$$, we have   \[I_k ( {{{\bf{{m}}}}}), k \frac{\partial I_k ( {{{\bf{{m}}}}})}{\partial k} = O (J ( {{{\bf{{m}}}}}, Y)).\] (31) To assist in the assessment of later sums whose terms are affected by factors of the type $$I_k ( {{{\bf{{m}}}}})$$ and $$k {\partial {^{{}}}{I_k ( {{{\bf{{m}}}}})} / \partial {{k}^{{}}}}$$ it is convenient to state at once the following: Lemma 5 Suppose$$X,Y,$$and$$N$$are as before and that$$ {{\mathrm {{q}}}} <X^{A}$$. Let$$G( {{{\bf{{m}}}}}, {{\mathrm {{q}}}}) \geq 0$$and suppose that for all$$y >0$$it is given that  \[\Gamma (y, {{\mathrm {{q}}}}) = \sum _{{0 <||{ {{{\bf{{m}}}}} }|| \leq y}} G( {{{\bf{{m}}}}}, {{\mathrm {{q}}}}) < \left({D_1( {{\mathrm {{q}}}}) +D_2( {{\mathrm {{q}}}}) y^{n_1}}\right) \log ^{n_2} (y {{\mathrm {{q}}}} +2),\]where$$D_1( {{\mathrm {{q}}}}), D_2( {{\mathrm {{q}}}}) >0$$and$$ {({n_1}, {n_2})} = {({n}, {0})}$$or$$ {({n - \frac {5}{2}}, {1})}$$. Then  \[\begin {array}{rl} \sum _{ { {{{\bf{{m}}}}} \neq 0}} ||{ {{{\bf{{m}}}}} }||^{\epsilon } G( {{{\bf{{m}}}}}, {{\mathrm {{q}}}}) J ( {{{\bf{{m}}}}}, Y) & = {O \left({{\frac { {{\min}^{{({1}/{2}) n - 1}}} (X, Y) X^{\epsilon } D_1( {{\mathrm {{q}}}})}{X^{({1}/{2}) n - 1}}}}\right)} \\ & \quad + {O \left({{\frac {Y^{({1}/{2}) n - 1} X^{\epsilon } N^{n_1 - ({1}/{2}) n +1} D_2( {{\mathrm {{q}}}})}{X^{n_1}}}}\right)}. \end {array}\] This is verified by a tedious but familiar process whereby the sum is split up into segments wherein $$ {{{\bf{{m}}}}}$$ is subject to a condition of the type $$2^{a} U \leq ||{ {{{\bf{{m}}}}} }|| <2^{a+1} U$$; here, $$U = 1$$ if $$Y \leq X$$ but $$U = Y / X$$ if $$Y >X$$. 6. The sum $$Q( {{{\bf{{m}}}}}, k)$$: first section The trigonometrical sum $$Q( {{{\bf{{m}}}}}, k)$$ defined by (9) is of the utmost importance to the investigation because of its presence in the fundamental inequality (14). It is the same as the sum $$Q_0 ( {{{\bf{{m}}}}}, k)$$ in Ib and Ic (or $$S_0 ( {{{\bf{{m}}}}}, k)$$ in Ha) and becomes the sum (35) in our Ia when $$n=6$$ and $$f ( {{{\bf{{x}}}}}) = g ( {{{\bf{{x}}}}}) = x_1^3 +x_2^3 +x_3^3 - x_4^3 - x_5^3 - x_6^3$$. We may therefore often speed our treatment either by citing results from these papers or by abbreviating some of the procedures therefrom. Yet our problems require us to delve much deeper into the properties of these sums than hitherto, particularly in the case where the modulus $$k$$ contains prime power factors with moderately large exponents. Estimates for both individual sums and sums of their moduli will be needed. Leaving the latter till later, we sketch the development of the theory of these sums until we reach the point when new results are to be considered. In doing this we shall involve the entities $$\nu (k)$$ and $$\nu ( {{{\bf{{m}}}}}, k)$$ that are defined, respectively, as the number of incongruent solutions, $$ {\hbox{mod}\, {k}}$$, of the congruence $$f( {{{\bf{{l}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {k}}$$, and the simultaneous congruences $$f( {{{\bf{{l}}}}}) \equiv {{{\bf{{m}}}}} {{{\bf{{l}}}}} \equiv 0$$, $$ {\hbox{mod}\, {k}}$$; associated with $$\nu (k)$$ is $$\nu ^{\prime }(k)$$, which is the number of primitive incongruent solutions, $$ {\hbox{mod}\, {k}}$$, of $$f( {{{\bf{{l}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {k}}$$, or, in other words, those for which $$ {({ {{{\bf{{l}}}}} }, {k})} = 1$$. Taken from §10 of Ib, where the functional symbol $$\rho$$ is used instead of $$\nu$$ here, there are then the useful estimates   \[\nu ^{\prime } (p^{\alpha }) = p^{(n-1)(\alpha - 1)} \nu ^{\prime } (p) \quad (p \nmid D), \quad \nu (p^{\alpha }) = O(p^{(n-1) \alpha })\] (32) that we shall need. The next result, which means we can often restrict attention to the case where $$k$$ is a prime power $$p^{\alpha }$$, is of a familiar type and can be proved, for example, in the same way as Lemma 5 in Ia (which relates to the special case where $$f( {{{\bf{{x}}}}})$$ is the form $$g( {{{\bf{{x}}}}})$$ above). Lemma 6 The sum$$Q ( {{{\bf{{m}}}}}, k)$$is a$$($$properly$$)$$multiplicative function of$$k,$$viz, if$$ {({k^{\prime }}, {k^{\prime \prime }})} = 1,$$then$$Q ( {{{\bf{{m}}}}}, k^{\prime } k^{\prime \prime }) = Q ( {{{\bf{{m}}}}}, k^{\prime }) Q ( {{{\bf{{m}}}}}, k^{\prime \prime })$$. As in §5 of Ia, the method starts to unfold through the equation   \[\begin {array}{rl} Q( {{{\bf{{m}}}}}, p^{\alpha }) & = \sum _{0 <{{{\bf{{l}}}}} \leq p^{\alpha }} \left({{\sum _{0 <h \leq p^{\alpha }}} e^{2 \pi i (h f( {{{\bf{{l}}}}}) - {{{\bf{{m}}}}} {{{\bf{{l}}}}}) / p^{\alpha }} - \sum _{{0 <h^{\prime } \leq p^{\alpha -1}}} e^{2 \pi i (p h^{\prime } f( {{{\bf{{l}}}}}) - {{{\bf{{m}}}}} {{{\bf{{l}}}}}) / p^{\alpha }}}\right) \\ & = p^{\alpha } \sum _{{\substack {0 <{{{\bf{{l}}}}} \leq p^{\alpha } \\ f( {{{\bf{{l}}}}}) \equiv 0, \, {\hbox{mod}\, {p^{\alpha }}} }}} e^{2 \pi i {{{\bf{{m}}}}} {{{\bf{{l}}}}} / p^{\alpha }} - p^{\alpha -1} \sum _{{\substack {0 <{{{\bf{{l}}}}} \leq p^{\alpha } \\ f( {{{\bf{{l}}}}}) \equiv 0, \, {\hbox{mod}\, {p^{\alpha -1}}} }}} e^{2 \pi i {{{\bf{{m}}}}} {{{\bf{{l}}}}} / p^{\alpha }} \\ & = p^{\alpha } Q_1 ( {{{\bf{{m}}}}}, p^{\alpha }) - p^{\alpha -1} Q_2 ( {{{\bf{{m}}}}}, p^{\alpha }) \quad {\mathrm {say}}. \end {array}\] (33) But, since in the sum defining $$Q_2 ( {{{\bf{{m}}}}}, p^{\alpha })$$ we may write $$ {{{\bf{{l}}}}} = {{{\bf{{l}}}}} ^{\prime } + {{{\bf{{r}}}}} p^{\alpha -1}$$ where $$f( {{{\bf{{l}}}}} ^{\prime }) \equiv 0$$, $$ {\hbox{mod}\, {p^{\alpha -1}}}$$, and $$0 <{{{\bf{{r}}}}} \leq p$$, we have   \[Q_2 ( {{{\bf{{m}}}}}, p^{\alpha }) = \sum _{{\substack {0 <{{{\bf{{l}}}}} ^{\prime } \leq p^{\alpha -1} \\ f( {{{\bf{{l}}}}} ^{\prime }) \equiv 0, \, {\hbox{mod}\, {p^{\alpha -1}}} }}} e^{2 \pi i {{{\bf{{m}}}}} {{{\bf{{l}}}}} ^{\prime } / p^{\alpha }} \sum _{{0 <{{{\bf{{r}}}}} \leq p}} e^{2 \pi i {{{\bf{{m}}}}} {{{\bf{{r}}}}} / p}\] in which the inner sum is $$0$$ or $$p^n$$ according as $$ {{{\bf{{m}}}}} {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p}}$$, or $$ {{{\bf{{m}}}}} \equiv 0$$, $$ {\hbox{mod}\, {p}}$$. Hence, if $$p \nmid {{{\bf{{m}}}}}$$, $$Q( {{{\bf{{m}}}}}, p^{\alpha })$$ is expressible through $$Q_1 ( {{{\bf{{m}}}}}, p^{\alpha })$$ alone by   \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = p^{\alpha } Q_1 ( {{{\bf{{m}}}}}, p^{\alpha }),\] (34) but is otherwise   \[\begin {array}{rl} & p^{\alpha } \sum _{{\substack {0 <{{{\bf{{l}}}}} \leq p^{\alpha } \\ f( {{{\bf{{l}}}}}) \equiv 0, \, {\hbox{mod}\, {p^{\alpha }}} }}} e^{2 \pi i {{{\bf{{m}}}}}_1 {{{\bf{{l}}}}} / p^{\alpha -1}} - p^{n + \alpha -1} \sum _{{\substack {0 <{{{\bf{{l}}}}} \leq p^{\alpha -1} \\ f( {{{\bf{{l}}}}}) \equiv 0, \, {\hbox{mod}\, {p^{\alpha -1}}} }}} e^{2 \pi i {{{\bf{{m}}}}}_1 {{{\bf{{l}}}}} / p^{\alpha -1}} \\ & \quad = p^{\alpha } Q_1^{ * } ( {{{\bf{{m}}}}}_1, p^{\alpha }) - p^{n + \alpha -1} Q_2^{ * } ( {{{\bf{{m}}}}}_1, p^{\alpha -1}) \quad {\mathrm {say}}, \end {array}\] (35) when $$ {{{\bf{{m}}}}} = p {{{\bf{{m}}}}}_1$$. The case where $$p \nmid {{{\bf{{m}}}}}$$ is the more important and presents extra difficulties for $$\alpha \geq 3$$ because then the size of $$Q_1 ( {{{\bf{{m}}}}}, p^{\alpha })$$ depends more significantly than for $$\alpha = 1$$, $$2$$ on the relationship of $$ {{{\bf{{m}}}}}$$ with the dual variety of $$ {{\cal {{{V}}}}}_p$$. The case $$p {{\vert }} {{{\bf{{m}}}}}$$ must also be treated because what we need is not available in earlier papers. We briefly summarize what we know about the sums $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ for $$\alpha = 1$$ or $$2$$ before going on to larger $$\alpha$$. First, stressing that (34) is valid for $$ {{{\bf{{m}}}}} {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p}}$$, even when $$F( {{{\bf{{m}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$, we note that, since the only property of $$g( {{{\bf{{x}}}}})$$ used in the proof of Lemma 6 of Ia was its homogeneity, we can confirm the truth of the following: Lemma 7 If$$ {{{\bf{{m}}}}} {{\not\equiv }} 0,$$$$ {\hbox{mod}\, {p}},$$then  \[Q( {{{\bf{{m}}}}}, p) = \frac {p}{p-1} {\{ } p \nu ( {{{\bf{{m}}}}}, p) - \nu (p) {\} }.\] The main application of this lemma must await our work on the Hasse–Weil zeta functions in §12, in which incidentally we could use it to establish the next lemma. Yet, in the meantime we appeal to Lemma 13 of Ha that is established by an alternative treatment. Lemma 8 We have  \[Q( {{{\bf{{m}}}}},p) = O(p^{({1}/{2}) n + {1}/{2}} (F( {{{\bf{{m}}}}}), p)^{{1}/{2}}).\] Actually, in the form stated, this assessment will only be needed when $$F( {{{\bf{{m}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$. A parallel result for the modulus $$p^2$$ is provided by Lemma 11 of Ib, which we state as follows: Lemma 9 We have  \[Q( {{{\bf{{m}}}}},p^2) = O(p^{n +1} (F( {{{\bf{{m}}}}}), p)).\] We should note that the better estimate supplied by Lemma 10 for the case $$\alpha =2$$ will not be needed. Having dealt with the more straightforward cases when $$\alpha \leq 2$$, we embark on the lengthy analysis involving higher powers of $$p$$. We begin, as in (37) of Ia for $$\alpha >1$$ and $$p \nmid {{{\bf{{m}}}}}$$, with the equation   \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = \frac {p^{\alpha }}{\phi (p^{\alpha })} {\{ } p^{\alpha } \nu _1 ( {{{\bf{{m}}}}}, p^{\alpha }) - p^{\alpha -1} \nu _2 ( {{{\bf{{m}}}}}, p^{\alpha }) {\} },\] where $$\nu _1 ( {{{\bf{{m}}}}}, p^{\alpha })$$ and $$\nu _2 ( {{{\bf{{m}}}}}, p^{\alpha -1})$$ are, respectively, the number of incongruent solutions, $$ {\hbox{mod}\, {p^{\alpha }}}$$, of the simultaneous conditions   \[f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {p^{\alpha }}}; \quad {{{\bf{{m}}}}} {{{\bf{{l}}}}} \equiv 0, {\hbox{mod}\, {p^{\alpha }}}; \quad {{{\bf{{l}}}}} {{\not\equiv }} 0, {\hbox{mod}\, {p}}\] (36)α and   \[f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {p^{\alpha }}}; \quad {{{\bf{{m}}}}} {{{\bf{{l}}}}} \equiv 0, {\hbox{mod}\, {p^{\alpha -1}}}; \quad {{{\bf{{l}}}}} {{\not\equiv }} 0, {\hbox{mod}\, {p}}.\] (37)α Next, departing slightly from the procedure in Ia, let us divide each sum $$\nu _i ( {{{\bf{{m}}}}}, p^{\alpha })$$ into the sums $$\nu _i^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha })$$, $$\nu _i^{ \mathrm {{(B)}} } ( {{{\bf{{m}}}}}, p^{\alpha })$$ by imposing the extra restriction that $$ {{{\bf{{l}}}}}$$ correspond, respectively, to a non-singular or singular point of $$ {{\cal {{{V}}}}}_p^{ {\dagger } } ( {{{\bf{{m}}}}})$$, viz. that $${\mathrm {grad}} f( {{{\bf{{l}}}}})$$ be non-proportional, $$ {\hbox{mod}\, {p}}$$, or (non-trivially) proportional, $$ {\hbox{mod}\, {p}}$$, to $$ {{{\bf{{m}}}}}$$. Then, if we compare $$\nu _1^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha })$$ and $$\nu _2^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha })$$ with $$\nu _1 ( {{{\bf{{m}}}}}, p^{\alpha -1})$$ by the method used to establish Lemma 7 in Ia, we find that   \[\nu _2^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha }) = p^{\alpha -1} \nu _1^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha -1}), \quad \nu _1^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha }) = p^{\alpha -2} \nu _1^{ \mathrm {{(A)}} } ( {{{\bf{{m}}}}}, p^{\alpha -1}) \] (38) with the consequence that   \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = \frac {p}{p-1} {\{ } p^{\alpha } \nu _1^{ \mathrm {{(B)}} } ( {{{\bf{{m}}}}}, p^{\alpha }) - p^{\alpha -1} \nu _2^{ \mathrm {{(B)}} } ( {{{\bf{{m}}}}}, p^{\alpha })) {\} }.\] (39) In particular, we have the following: Lemma 10 For$$\alpha >1$$and$$F( {{{\bf{{m}}}}}) {{\not\equiv }} 0,$$$$ {\hbox{mod}\, {p}},$$  \[Q( {{{\bf{{m}}}}}, p^{\alpha }) = 0.\] This completes our initial discussion, in which it must be observed that there is no restriction on $$p$$. Yet when we go on to derive further estimates for $$Q( {{{\bf{{m}}}}}, p^{\alpha })$$ for various values of $$\alpha$$ exceeding $$2$$ we shall often have to insist that $$p >p_0$$, a matter normally of little importance because in the opposite case the sum is bounded provided $$\alpha$$ be limited by a number such as $$6$$. 7. The sums $$Q( {{{\bf{{m}}}}}, p^{\alpha })$$ for $$p \nmid {{{\bf{{m}}}}};$$ the case $$\alpha =3$$ For the case $$\alpha = 3,$$ we shall require the following: Lemma 11 Let$$q ( {{x}_{1}, \dotsc , {x}_{{M}}})$$and$$l ( {{x}_{1}, \dotsc , {x}_{{M}}})$$be a quadratic and a linear form over$${\Bbb {F}}_p$$and suppose the rank of the symmetric matrix appertaining to$$q ( {{x}_{1}, \dotsc , {x}_{{M}}})$$is$$r$$. Then the number$$w(p)$$of solutions over$${\Bbb {F}}_p$$of the equation  \[\phi ( {{x}_{1}, \dotsc , {x}_{{M}}}) = q ( {{x}_{1}, \dotsc , {x}_{{M}}}) +l ( {{x}_{1}, \dotsc , {x}_{{M}}}) +m = 0\]is equal to  \[p^{M-1} +O(p^{M - ({1}/{2}) r}).\] This estimate, which alternatively can be expressed in the language of congruences, can be derived after several transformations of the unknowns by means of estimates for Gaussian sums. Yet it is swifter to avail ourselves of the theory of equations over finite fields, the relevant portion of which can be found, for example, in our article [11]. The case $$r \leq 2$$ being trivial because $$w(p) = O(p^{M-1})$$ or $$O(p^{M})$$ according as $$q( {{x}_{1}, \dotsc , {x}_{{n}}})$$ is non-zero or zero, we first form in the opposite instance the homogeneous completion   \[\psi ( {{x}_{1}, \dotsc , {x}_{{M+1}}}) = q( {{x}_{1}, \dotsc , {x}_{{M}}}) +x_{M+1} l( {{x}_{1}, \dotsc , {x}_{{M}}}) +m x_{M+1}^{2}\] of $$\phi ( {{x}_{1}, \dotsc , {x}_{{M}}})$$ and, using the language of [11, §2], consider the connection between the (projective) dimensions $$d_1$$, $$d_2$$ of the singular loci of the equations$$\psi ( {{x}_{1}, \dotsc , {x}_{{M+1}}}) = 0$$ and $$q( {{x}_{1}, \dotsc , {x}_{{M}}}) = 0$$. The latter equation being formed by setting $$x_{M+1} = 0$$ in the former we find by the reasoning in [11] that $$d_1 \leq d_2 +1$$ and hence that   \[d_1 \leq M - r \leq M - 2,\] (40) since the equality $$d_2 = M - r - 1$$ follows from the identity of the matrices associated with $$q( {{x}_{1}, \dotsc , {x}_{{M}}})$$ and $${\mathrm {grad}} q( {{x}_{1}, \dotsc , {x}_{{M}}})$$. From (40) it follows that $$d_1$$ is actually the dimension of the singular locus of the projective hypersurface $$\psi ( {{x}_{1}, \dotsc , {x}_{{M+1}}}) = 0$$, and we deduce from [11, §5] that   \[w(p) = p^{M-1} +O({p^{(M - 1 +M - r +1)/2}}) = p^{M-1} +O({p^{M - ({1}/{2})r}}),\] as asserted. Before advancing further, we remark that in the estimation of $$Q( {{{\bf{{m}}}}}, p^{\alpha })$$ when $$p \nmid {{{\bf{{m}}}}}$$ we can actually assume that   \[( {{m}_{1}, \dotsc , {m}_{{n}}}) = 1\] (41) because it is easily seen from (9) that $$Q (\lambda {{{\bf{{m}}}}}, p^{\alpha }) = Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ when $$p \nmid \lambda$$ and hence when $$\lambda \equiv \overline {( {{m}_{1}, \dotsc , {m}_{{n}}})}$$, $$ {\hbox{mod}\, {p}}$$. In treating $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ by (39) for $$p \nmid {{{\bf{{m}}}}}$$ and $$p >p_0$$, we shall throughout omit the superscript from $$\nu _1^{ \mathrm {{(B)}} } ( {{{\bf{{m}}}}}, p^{\alpha })$$ and $$\nu _2^{ \mathrm {{(B)}} } ( {{{\bf{{m}}}}}, p^{\alpha })$$ for ease of notation and assume, as we may, that $$F( {{{\bf{{m}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$. Also, by a singular solution of   \[f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {p}}; \quad {{{\bf{{m}}}}} {{{\bf{{l}}}}} \equiv 0, {\hbox{mod}\, {p}},\] (42) we mean in an affine sense any vector $$ {{{\bf{{l}}}}} = ( {{l}_{1}, \dotsc , {l}_{{n}}}) \neq 0$$ that represents projectively a singularity of $$ {{\cal {{{V}}}}}_p ( {{{\bf{{m}}}}})$$. Thus, in defining $$\nu _1 ( {{{\bf{{m}}}}}, p^{\alpha })$$ and $$\nu _2 ( {{{\bf{{m}}}}}, p^{\alpha })$$ with their new meaning we use the sets of conditions (36)$$_{\alpha }^{\prime }$$ and (37)$$_{\alpha }^{\prime }$$ that are formed by adding to (36)$$_{\alpha }$$ and (37)$$_{\alpha }$$ the condition that $$ {{{\bf{{l}}}}}$$ be a singular solution of (42). Each solution of (36)$$_{3}^{\prime }$$ and (37)$$_{3}^{\prime }$$ stems in an obvious manner from a solution $$ {{{\bf{{l}}}}}_2$$, $$ {\hbox{mod}\, {p^2}}$$, of (36)$$_{2}^{\prime }$$ that in turn is derived from a singular solution $$ {{{\bf{{l}}}}}_1$$ of (42). To assess how many incongruent solutions $$ {{{\bf{{l}}}}}_2$$, $$ {\hbox{mod}\, {p^2}}$$ originate from a given $$ {{{\bf{{l}}}}}_1$$ we express $$ {{{\bf{{l}}}}}_2$$ as $$ {{{\bf{{l}}}}}_1 + {{{\bf{{r}}}}} p$$, where $$0 <{{{\bf{{r}}}}} \leq p$$, and obtain the conditions   \[\begin {array}{rl} f( {{{\bf{{l}}}}}_2) & \equiv f( {{{\bf{{l}}}}}_1) +p {\mathrm {grad}} f( {{{\bf{{l}}}}}_1). {{{\bf{{r}}}}} \equiv 0, {\hbox{mod}\, {p^2}}, \\ {{{\bf{{m}}}}} {{{\bf{{l}}}}}_2 & \equiv {{{\bf{{m}}}}} {{{\bf{{l}}}}}_1 +p {{{\bf{{m}}}}} {{{\bf{{r}}}}} \equiv 0, {\hbox{mod}\, {p^2}}, \end {array}\] which are tantamount to   \[\begin {array}{rl} {\mathrm {grad}} f( {{{\bf{{l}}}}}_1). {{{\bf{{r}}}}} & \equiv - f( {{{\bf{{l}}}}}_1)/p, {\hbox{mod}\, {p}}, \\ {{{\bf{{m}}}}} {{{\bf{{r}}}}} & \equiv - {{{\bf{{m}}}}} {{{\bf{{l}}}}}_{1}/p, {\hbox{mod}\, {p}}. \end {array}\] Since $$ {{{\bf{{m}}}}}$$ and $${\mathrm {grad}} f( {{{\bf{{l}}}}}_1)$$ are proportional, $$ {\hbox{mod}\, {p}}$$, there are therefore either $$p^{n-1}$$ solutions in $$ {{{\bf{{r}}}}}$$ or none, and we therefore confine attention to the $$ {{{\bf{{l}}}}}_1$$ that give rise to the former situation. Let us consider the contribution to $$\nu _2 ( {{{\bf{{m}}}}}, p^3)$$ that is created by one of the vectors $$ {{{\bf{{l}}}}}_2$$ above. This, by the very argument used to get the second part of (38), is $$p^{n-1}$$ since $$p \nmid D$$ and therefore the effect of a non-excluded $$ {{{\bf{{l}}}}}_1$$ on $$\nu _2 ( {{{\bf{{m}}}}}, p^3)$$ is   \[p^{2n - 2}.\] (43) The estimation of $$\nu _1 ( {{{\bf{{m}}}}}, p^3)$$ necessitates a transformation of (36)$$_{\alpha }^{\prime }$$. Since $$ {{h.c.f.}({ {{m}_{1}, \dotsc , {m}_{{n}}} })} = 1$$ by the legitimate assumption (41), there is a unimodular matrix $$ {{{\bf{{P}}}}}$$ with rational integral constituents having $$ {{m}_{1}, \dotsc , {m}_{{n}}}$$ as the first row, by which we form the substitution   \[ {{{\bf{{P}}}}} {{{\bf{{x}}}}} = {{{\bf{{X}}}}} ^{ {\dagger } }\] (44) that throws $$f( {{{\bf{{x}}}}})$$ into a polynomial $$h( {{{\bf{{X}}}}} ^{ {\dagger } })$$ and $$ {{{\bf{{m}}}}} {{{\bf{{x}}}}}$$ into $$X_{1}^{ {\dagger } }$$. To utilize this let (Not to be confused with the Hessian $$H( {{{\bf{{x}}}}})$$ for which italic font is used.) $${\mathrm {H}} ( {{{\bf{{X}}}}}) = {\mathrm {H}} (X_2, \ldots , X_n) = h (0, X_2, \ldots , X_n)$$ and let $$(L_1^{ {\dagger } }, L_2, \ldots , L_n)$$ be the transform of $$( {{l}_{1}, \dotsc , {l}_{{n}}})$$, then writing $$ {{{\bf{{L}}}}} = (L_2, \ldots , L_n)$$ so that, moreover, $$ {{{\bf{{L}}}}}_i$$ is to correspond to any vector designated by $$ {{{\bf{{l}}}}}_i$$ in our present or future processes. Then the equations (36)$$_{\alpha }^{\prime }$$ are equivalent to the triplet of conditions   \[{\mathrm {H}} ( {{{\bf{{L}}}}}) \equiv 0, {\hbox{mod}\, {p^{\alpha }}}, \quad {{{\bf{{L}}}}} {{\not\equiv }} 0, {\hbox{mod}\, {p}}, \quad L_1^{ {\dagger } } \equiv 0, {\hbox{mod}\, {p^{\alpha }}},\] the last of which may obviously be ignored by simply assigning the value zero to $$L_1^{ {\dagger } }$$ throughout. As explained in § 3 the number of projective points represented by singular solutions $$ {{{\bf{{l}}}}}_1$$ of (42) is $$O(1)$$ for a given vector $$ {{{\bf{{m}}}}}$$ for which $$F( {{{\bf{{m}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$, and $$ {{{\bf{{m}}}}} {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p}}$$. Let (Because $$ {{{\bf{{M}}}}} ( {{{\bf{{l}}}}}_1) = 0$$ would imply that $${\mathrm {grad}} f( {{{\bf{{l}}}}}_1) = 0$$. We shall later see that in fact $$r \geq 2$$ but this is immaterial until we later discuss the significance of the number $$r$$.) $$r \geq 1$$ be the minimal value of the rank, $$ {\hbox{mod}\, {p}}$$, of the Hessian matrix $$ {{{\bf{{M}}}}} ( {{{\bf{{l}}}}}_1)$$ for those $$ {{{\bf{{l}}}}}_1$$ that produce a contribution to $$\nu _1 ( {{{\bf{{m}}}}}, p^2)$$. Then, translating into the language of the transformed coordinates, we need to count the zeros of $${\mathrm {H}} ( {{{\bf{{L}}}}})$$, $$ {\hbox{mod}\, {p^3}}$$, that are congruent to a given valid $$ {{{\bf{{L}}}}}_1$$, $$ {\hbox{mod}\, {p}}$$, which we have already shown to have the property that actually $${\mathrm {H}} ( {{{\bf{{L}}}}}_1) \equiv 0$$, $$ {\hbox{mod}\, {p^2}},$$ because of the way $$ {{{\bf{{l}}}}}_1$$ was chosen. Thus, writing (Note that $$ {{{\bf{{s}}}}}$$ has $$n-1$$ components.)   \[ {{{\bf{{L}}}}}_3 = {{{\bf{{L}}}}}_1 +p {{{\bf{{s}}}}} \quad {\mathrm {for\ }} 0 <{{{\bf{{s}}}}} \leq p^2\] (45) and designating the Hessian matrix of $${\mathrm {H}} ( {{{\bf{{L}}}}})$$ by $$ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}})$$, we require that (Note $$\frac {1}{2} {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1)$$ has integral constituents. Note also the comment in §2.)   \[{\mathrm {H}} ( {{{\bf{{L}}}}}_3) \equiv {\mathrm {H}} ( {{{\bf{{L}}}}}_1) +p {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1). {{{\bf{{s}}}}} + \frac {1}{2} p^2 {{{\bf{{s}}}}} ^{\prime } {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} \equiv 0, {\hbox{mod}\, {p^3}},\] in which $${\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$, because $$ {{{\bf{{L}}}}}_1$$ is a singularity of the hypersurface $${\mathrm {H}} ( {{{\bf{{X}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$. Hence,   \[ {{{\bf{{s}}}}} ^{\prime } {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} +2 ({\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) / p). {{{\bf{{s}}}}} +2 {\mathrm {H}} ( {{{\bf{{L}}}}}_1) / p^2 \equiv 0, {\hbox{mod}\, {p}}.\] (46) To exploit this congruence through Lemma 11, we must consider the rank $$r_1$$ of $$ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1)$$, $$ {\hbox{mod}\, {p}}$$. This matrix is obtained by removing the first row and first column of the Hessian matrix $$ {{{\bf{{M}}}}} ( {{{\bf{{L}}}}}_1^{ {\dagger } })$$, which, being the transform of $$ {{{\bf{{M}}}}} ( {{{\bf{{l}}}}}_1)$$ under the substitution (44), has rank not less than $$r$$. Hence, $$r_1 \geq r - 2$$ and $$r_1 \geq 0$$. Consequently, by Lemma 11, the number of solutions of (46) subject to the inequality in (45) is   \[p^{n-1} {\{ } p^{n-2} +O(p^{n-1-({1}/{2}) r_1}) {\} } = p^{2n-3} +O(p^{2n-1-({1}/{2})r}).\] We go back to (39), which shows through (43) and the above equality that the effect of a single eligible solution on $$Q( {{{\bf{{m}}}}}, p^3)$$ is   \[\frac {p}{p-1} (p^{2n} +O(p^{2n+2-({1}/{2})r}) - p^{2n}) = O(p^{2n+2-({1}/{2})r}).\] Since each qualifying projective solution of (42) gives rise to $$p-1$$ incongruent affine counterparts and since $$ {{\cal {{{V}}}}}_p ( {{{\bf{{m}}}}})$$ has a bounded number of projective singularities, we deduce the following: Lemma 12 Suppose$$p >p_0$$. Then, if$$F( {{{\bf{{m}}}}}) \equiv 0,$$$$ {\hbox{mod}\, {p}},$$and$$r$$be defined as above, we have  \[Q ( {{{\bf{{m}}}}}, p^3) = O (p^{2n+3-({1}/{2})r}).\] Alternatively, it is clear that $$Q ( {{{\bf{{m}}}}}, p^3) = 0$$ when $$r$$ is not defined, namely, when no $$ {{{\bf{{l}}}}}_1$$ gives rise to an $$ {{{\bf{{l}}}}}_2$$. 8. The sums $$Q( {{{\bf{{m}}}}}, p^{\alpha })$$ for $$p \nmid {{{\bf{{m}}}}};$$ the case $$\alpha \geq 4$$ When $$\alpha$$ goes beyond $$3$$ the method of estimating $$Q( {{{\bf{{m}}}}}, p^{\alpha })$$ changes although we extend the structure of the zeros $$ {{{\bf{{L}}}}}_{\alpha }$$, $$ {\hbox{mod}\, {p^{\alpha }}}$$, of $${\mathrm {H}} ( {{{\bf{{X}}}}})$$ that stem successfully from an eligible zero $$ {{{\bf{{L}}}}}_1$$, $$ {\hbox{mod}\, {p}}$$, by way of eligible zeros $$ {{{\bf{{L}}}}}_2$$, $$ {\hbox{mod}\, {p^2}}, \ldots$$, to $$ {{{\bf{{L}}}}}_{\alpha -1}$$, $$ {\hbox{mod}\, {p^{\alpha -1}}}$$, where $$ {{{\bf{{L}}}}}_i \equiv {{{\bf{{L}}}}}_{i-1}$$, $$ {\hbox{mod}\, {p^{i-1}}}$$. If a zero $$ {{{\bf{{L}}}}}_{\alpha -1}$$ stem from $$ {{{\bf{{L}}}}}_{\alpha -2}$$, then for some $$ {{{\bf{{s}}}}}$$ we would have   \[{\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2} + {{{\bf{{s}}}}} p^{\alpha -2}) \equiv {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) +p^{\alpha -2} {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}). {{{\bf{{s}}}}} \equiv {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) \equiv 0, {\hbox{mod}\, {p^{\alpha -1}}},\] the requirement of which is met either always or never; in the former case $$ {{{\bf{{L}}}}}_{\alpha -2}$$ is itself a vector $$ {{{\bf{{L}}}}}_{\alpha -1}$$ and gives rise to $$p^{n-1}$$ vectors, incongruent to the modulus $$p^{\alpha -1}$$, of this type, whereas in the latter case we ignore the vector $$ {{{\bf{{L}}}}}_{\alpha -2}$$ because it is not the forbear of an $$ {{{\bf{{L}}}}}_{\alpha -1}$$. Then the contribution to $$\nu _1 ( {{{\bf{{m}}}}}, p^{\alpha -1})$$ derived from a single eligible $$ {{{\bf{{L}}}}}_{\alpha -2}$$ via its offspring $$ {{{\bf{{L}}}}}_{\alpha -1}$$ is $$p^{n-1}$$, whence by a previous argument the influence of this $$ {{{\bf{{L}}}}}_{\alpha -2}$$ on $$\nu _2 ( {{{\bf{{m}}}}}, p^{\alpha })$$ is   \[p^{n-1} p^{n-1} = p^{2n-2}.\] (47) To think about the contribution of the above eligible vector $$ {{{\bf{{L}}}}}_{\alpha -2}$$ to $$\nu _1 ( {{{\bf{{m}}}}}, p^{\alpha })$$ by way of the vectors $$ {{{\bf{{L}}}}}_{\alpha -2} + {{{\bf{{s}}}}} p^{\alpha -2}$$, we must form the conditions   \[{\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) +p^{\alpha -2} {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}). {{{\bf{{s}}}}} \equiv 0, {\hbox{mod}\, {p^{\alpha }}}, \quad 0 <{{{\bf{{s}}}}} \leq p^2,\] (48) because $$2 (\alpha -2) \geq \alpha$$ when $$\alpha >3$$. There thus results the congruence   \[({\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) / p). {{{\bf{{s}}}}} \equiv - {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) / p^{\alpha -1}, {\hbox{mod}\, {p}},\] that has $$p^{n-2}$$ incongruent solutions in $$ {{{\bf{{s}}}}}$$, $$ {\hbox{mod}\, {p}}$$, unless $${\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) \equiv 0$$, $$ {\hbox{mod}\, {p^2}}$$, in which event it has $$p^{n-1}$$ or no solutions. In the earlier situation the number of solutions of (48) is $$p^{2n-3}$$ and we deduce from (39) and (47) that the impact of the zero $$ {{{\bf{{L}}}}}_{\alpha -2}$$ on $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ is   \[\frac {p}{p-1} (p^{2n-3 + \alpha } - p^{2n-3 + \alpha }) = 0.\] Yet in the other case it is obviously $$O (p^{2n-2 + \alpha }),$$ so that we need to know how many $$ {{{\bf{{L}}}}}_{\alpha -2}$$ can arise for which   \[{\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -2}) \equiv 0, {\hbox{mod}\, {p^2}}.\] (49) But each $$ {{{\bf{{L}}}}}_{\alpha -2}$$ is congruent, $$ {\hbox{mod}\, {p^2}}$$, to a vector $$ {{{\bf{{L}}}}}_2 = {{{\bf{{L}}}}}_1 +p {{{\bf{{s}}}}}$$ with $$0 <{{{\bf{{s}}}}} \leq p$$. We therefore require that   \[{\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{1}) +p {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} \equiv 0, {\hbox{mod}\, {p^2}},\] namely that   \[ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} \equiv - {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{1}) / p, {\hbox{mod}\, {p}},\] the number of whose solutions in $$ {{{\bf{{s}}}}}$$, $$ {\hbox{mod}\, {p}}$$, does not exceed   \[p^{n-1 - r_1} \leq p^{n+1 - r}.\] Consequently, the number of qualifying incongruent $$ {{{\bf{{L}}}}}_{\alpha -2}$$, $$ {\hbox{mod}\, {p^{\alpha -2}}}$$, satisfying (49) amounts to   \[O({p^{(n-1) (\alpha -4)} p^{n+1 - r}})\] and we altogether achieve the bound   \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = O({p^{2n-2 + \alpha + (n-1) (\alpha -4) +n+1 - r +1}}) = O({p^{n \alpha - n+4 - r}})\] because $$p-1$$ vectors $$ {{{\bf{{L}}}}}_1$$ arise from each singular point of $$ {{\cal {{{V}}}}}_p ( {{{\bf{{m}}}}})$$. We thus have the following lemma: Lemma 13 Under the conditions laid down in the statement of Lemma 12, we have  \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = O({p^{n \alpha - n+4 - r}})\]when$$\alpha \geq 4$$. A comment similar to that made after Lemma 12 is applicable. By the contravariance of the form $$F( {{{\bf{{u}}}}}),$$ we see through an examination of the substitution (44) that $$F( {{{\bf{{m}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p^2}}$$, if the congruence $${\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_2) \equiv 0$$, $$ {\hbox{mod}\, {p^2}}$$, be granted. However, the consequential improvement in the estimate in the lemma to zero when $$p {{\parallel } } F( {{{\bf{{m}}}}})$$ is of no tangible benefit in the calculations that follow. This lemma will only be used when $$\alpha = 4$$ and must be replaced by a more accurate result for values of $$\alpha$$ exceeding $$5$$, the case $$\alpha =5$$ being subsumed in another procedure. Clearly, we need only consider the case where (49) holds in revising the treatment of Lemma 13 when $$\alpha \geq 6$$. Each relevant vector $$ {{{\bf{{L}}}}}_{\alpha -2}$$ must then proceed from a vector $$ {{{\bf{{L}}}}}_{\alpha -3}$$ that has the property that   \[{\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) \equiv 0, {\hbox{mod}\, {p^2}},\] because $$\alpha \geq 5$$. Moreover, in determining how vectors $$ {{{\bf{{L}}}}}_{\alpha -1}$$ can proceed from $$ {{{\bf{{L}}}}}_{\alpha -3}$$, we set   \[ {{{\bf{{L}}}}}_{\alpha -1} = {{{\bf{{L}}}}}_{\alpha -3} +p^{\alpha -3} {{{\bf{{s}}}}}, \quad 0 <{{{\bf{{s}}}}} \leq p^2,\] and require that   \[{\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -1}) \equiv {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) +p^{\alpha -3} {{{\bf{{s}}}}}. {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) \equiv {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) \equiv 0, {\hbox{mod}\, {p^{\alpha -1}}}.\] There thus being no contribution to $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ via a vector $$ {{{\bf{{L}}}}}_{\alpha -3}$$ for which $${\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p^{\alpha -1}}}$$, we find in the contrary instance that the impact of $$ {{{\bf{{L}}}}}_{\alpha -3}$$ on $$\nu _2 ( {{{\bf{{m}}}}}, p^{\alpha })$$ is   \[p^{n-1} \cdot p^{2(n-1)} = p^{3(n-1)}\] (50) by the reasoning that led to (47). To find the $$ {{{\bf{{L}}}}}_{\alpha }$$ stemming from an eligible $$ {{{\bf{{L}}}}}_{\alpha -3}$$ and hence from an $$ {{{\bf{{L}}}}}_{\alpha -1}$$ as above we use the condition   \[{\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) +p^{\alpha -3} {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}). {{{\bf{{s}}}}} \equiv 0, {\hbox{mod}\, {p^{\alpha }}}, \quad 0 <{{{\bf{{s}}}}} \leq p^3,\] (51) that is valid for $$\alpha \geq 6$$, wherefore   \[{\{ } {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) / p^2 {\} }. {{{\bf{{s}}}}} \equiv - {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) / p^{\alpha -1}, {\hbox{mod}\, {p}},\] (52) which congruence has $$p^{n-2}$$ incongruent solutions, $$ {\hbox{mod}\, {p}}$$, when $${\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p^3}}$$. In this case (51) has $$p^{2n-2} p^{n-2} = p^{3n-4}$$ solutions, and we see from (50) and a previous argument that the contribution of $$ {{{\bf{{L}}}}}_{\alpha -3}$$ to $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ is zero. There remains for consideration the case where   \[{\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_{\alpha -3}) \equiv 0, {\hbox{mod}\, {p^3}},\] (53) and where therefore the effect of $$ {{{\bf{{L}}}}}_{\alpha -3}$$ on $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ is $$O(p^{3n-3 + \alpha })$$ because then (52) has either $$p^{n-1}$$ or $$0$$ incongruent solutions, $$ {\hbox{mod}\, {p}}$$. Hence, needing to learn how many $$ {{{\bf{{L}}}}}_{\alpha -3}$$ arise in this situation, we shall analyse the congruence (53) by a slightly altered version of the method used to treat (49). Since each such $$ {{{\bf{{L}}}}}_{\alpha -3}$$ is congruent, $$ {\hbox{mod}\, {p^3}}$$, to a vector   \[ {{{\bf{{L}}}}}_3 = {{{\bf{{L}}}}}_1 +p {{{\bf{{s}}}}}\] (54) with $$0 <{{{\bf{{s}}}}} \leq p^2$$, we require that $$ {{{\bf{{s}}}}}$$ satisfy the congruence   \[{\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) +p {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} +p^2 {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{s}}}}}) \equiv 0, {\hbox{mod}\, {p^3}},\] and hence the congruence   \[ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} +p {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{s}}}}}) \equiv - {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) /p, {\hbox{mod}\, {p^2}},\] (55) which in the first place implies that   \[ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}} \equiv - {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) /p, {\hbox{mod}\, {p}}.\] Consequently, $$ {{{\bf{{s}}}}}$$ belongs to at most $$p^{n+1-r}$$ residue classes $$ {{{\bf{{b}}}}}$$, $$ {\hbox{mod}\, {p}}$$, whence, setting $$ {{{\bf{{s}}}}} = {{{\bf{{b}}}}} +p {{{\bf{{s}}}}}_1$$ in (55), we find that   \[ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{b}}}}} +p {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}}_1 +p {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{b}}}}}) \equiv - ({\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) /p), {\hbox{mod}\, {p^2}},\] and thus that   \[ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{s}}}}}_1 \equiv - {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{b}}}}}) - \dfrac {1}{p} \left({ {{{\bf{{M}}}}} ^{ * } ( {{{\bf{{L}}}}}_1) {{{\bf{{b}}}}} + {\mathrm {grad}} {\mathrm {H}} ( {{{\bf{{L}}}}}_1) /p}\right), {\hbox{mod}\, {p}},\] the last congruence having at most $$p^{n+1 - r}$$ incongruent solutions in $$ {{{\bf{{s}}}}}_1$$, $$ {\hbox{mod}\, {p}}$$. Hence there are at most $$p^{2(n+1 - r)}$$ appropriate values of $$ {{{\bf{{s}}}}}$$ in (54) and the number of incongruent solutions, $$ {\hbox{mod}\, {p^{\alpha -3}}}$$, of (53) stemming from a given $$ {{{\bf{{L}}}}}_1$$ is   \[O({p^{(n-1) (\alpha -6)} p^{2 (n+1 - r)}}).\] In all, therefore, we conclude that   \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = O({p \cdot p^{3n-3 + \alpha } \cdot p^{(n-1) (\alpha -6)} \cdot p^{2 (n+1 - r)}}) = O({p^{n \alpha -n+6 - 2r}})\] with the result that we can state the following: Lemma 14 The exponent of$$p$$in the estimate for$$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$in Lemma 13 can be replaced by$$n \alpha -n+6 - 2r$$when$$\alpha \geq 6$$. To conclude our discussion of the case $$p \nmid {{{\bf{{m}}}}}$$ it is now appropriate to discuss the significance of the number $$r$$ appearing in the work when $$p >p_0$$. From its genesis, it is apparent that it depends on the affine vector $$ {{{\bf{{m}}}}}$$ so that it is appropriate to describe it as $$r ( {{{\bf{{m}}}}}, p)$$ when desirable; it is in fact determined as the rank of a Hessian matrix associated with a singular linear section of $$ {{\cal {{{V}}}}}_p$$ cut by the hyperplane $$ {{{\bf{{m}}}}} {{{\bf{{x}}}}} = 0$$. Also, in view of Lemma 2, $$r ( {{{\bf{{m}}}}}, p)$$ cannot assume the values $$0$$ or $$1$$ because the $$ {{{\bf{{l}}}}}_1$$ appearing in the definition of $$r$$ is incongruent to $$0$$, $$ {\hbox{mod}\, {p}}$$. Thus, if we interpret matters in terms of the field $${\Bbb {F}}_p$$, the estimates in Lemma 12– 14 are true for a given value of $$r \geq 2$$ when $$ {{{\bf{{m}}}}}$$ belongs to a certain set of points within an affine variety whose dimension $$\dim (r, p)$$ is determined by the requirement that $$r = r ( {{{\bf{{m}}}}}, p)$$. Indeed, by Lemma 2, this dimension is seen to be subject to the inequality   \[\dim (r, p) \leq \left\{ \begin {array}{ll}r - 1 & {\mathrm {if\ }} r = n-1 {\mathrm {\ or\ }} n, \\ r & {\mathrm {if\ }} 2 \leq r \leq n-2, \end {array}\right .\] (56) through the mapping $${\mathrm {grad}} \ {{\bf{f}}}( {{{\bf{{l}}}}}_1) \mapsto {{{\bf{{m}}}}}$$, where $$ {{{\bf{{l}}}}}_1$$ retains the meaning assigned to it above. Except when $$\alpha = 3$$ the weaker inequality $$\dim (r, p) \leq r$$ is all that we can effectively use in the sequel. The $$ {{{\bf{{m}}}}}$$ for which $$r$$ is not defined will be ignored in future calculations because in this case, as has already been observed, $$Q ( {{{\bf{{m}}}}}, p^{\alpha }) = 0$$. 9. The sums $$Q ( {{{\bf{{m}}}}}, p^{\alpha })$$ when $$p {{\vert }} {{{\bf{{m}}}}}$$ The estimation of these sums becomes complicated in this situation when the general case is considered. But here a simple treatment is possible because we only need assessments when $$\alpha = 3$$ or $$4$$, in which circumstances we need not take any account of any cancellations due to the presence of the complex exponentials in the sum. We begin with (35), writing   \[Q_1^{ * } ( {{{\bf{{m}}}}}_1, p^{\alpha }) = \sum _{p \nmid {{{\bf{{l}}}}} } + \sum _{p {\, {\vert } \,} {{{\bf{{l}}}}} } = Q_1^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha }) +Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^{\alpha })\] and   \[Q_2^{ * } ( {{{\bf{{m}}}}}_1, p^{\alpha -1}) = \sum _{p \nmid {{{\bf{{l}}}}} } + \sum _{p {\, {\vert } \,} {{{\bf{{l}}}}} } = Q_2^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha -1}) +Q_2^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^{\alpha -1}).\] Then, expressing the variable of summation $$ {{{\bf{{l}}}}}$$ in $$Q_1^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha })$$ as $$ {{{\bf{{l}}}}} ^{\prime } +p^{\alpha -1} {{{\bf{{u}}}}}$$ where $$p \nmid {{{\bf{{l}}}}} ^{\prime }$$, $$0 <{{{\bf{{l}}}}} ^{\prime } \leq p^{\alpha -1}$$, $$f( {{{\bf{{l}}}}} ^{\prime }) \equiv 0$$, $$ {\hbox{mod}\, {p^{\alpha -1}}}$$, and $$0 <{{{\bf{{u}}}}} \leq p$$, we demand that   \[f( {{{\bf{{l}}}}} ^{\prime } +p^{\alpha -1} {{{\bf{{u}}}}}) \equiv f( {{{\bf{{l}}}}} ^{\prime }) +p^{\alpha -1} {\mathrm {grad}} f( {{{\bf{{l}}}}} ^{\prime }). {{{\bf{{u}}}}} \equiv 0, {\hbox{mod}\, {p^{\alpha }}},\] and thus that   \[{\mathrm {grad}} f( {{{\bf{{l}}}}} ^{\prime }). {{{\bf{{u}}}}} \equiv - f( {{{\bf{{l}}}}} ^{\prime }) / p^{\alpha -1}, {\hbox{mod}\, {p}},\] which congruence has $$p^{n-1}$$ incongruent solutions, $$ {\hbox{mod}\, {p}}$$, in $$ {{{\bf{{u}}}}}$$ because $$ {{\cal {{{V}}}}}_p$$ is non-singular for $$p \nmid D$$. Since the summand in $$Q_1^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha })$$ is equal to $$e^{2 \pi i {{{\bf{{m}}}}}_1 {{{\bf{{l}}}}} ^{\prime } / p^{\alpha -1}}$$, we deduce that $$Q_1^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha }) = p^{n-1} Q_2^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha -1})$$ and hence that   \[p^{\alpha } Q_1^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha }) - p^{n+ \alpha -1} Q_2^{\prime } ( {{{\bf{{m}}}}}_1, p^{\alpha -1}) = 0.\] (57) Next, since $$ {{{\bf{{l}}}}}$$ is of the form $$p {{{\bf{{l}}}}}_1$$ in the conditions governing the sums $$Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^{\alpha })$$ and $$Q_2^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^{\alpha -1})$$, we see that these sums are taken over vectors $$ {{{\bf{{l}}}}}_1$$ that range, respectively, up to $$p^{\alpha -1}$$ and $$p^{\alpha -2}$$, as a consequence of which the sizes of $$Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^3)$$, $$Q_2^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^3)$$, and $$Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^2)$$ are $$p^{2n}$$, $$p^{2n}$$, and $$p^{n}$$. But, for $$Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^4)$$ we must use the condition $$f( {{{\bf{{l}}}}}_1) \equiv 0$$, $$ {\hbox{mod}\, {p}}$$, in addition to the inequality $$0 <{{{\bf{{l}}}}}_1 \leq p^3$$, whence   \[|{Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^4)}| \leq p^{2n} \nu (p) = O (p^{2n} \cdot p^{n-1}) = O (p^{3n-1})\] by (32). Consolidating these results in the statement   \[p^{\alpha } Q_1^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^{\alpha }) - p^{n + \alpha -1} Q_2^{\prime \prime } ( {{{\bf{{m}}}}}_1, p^{\alpha -1}) = \left\{ \begin {array}{ll}O (p^{2n+3}) & {\mathrm {if\ }} \alpha = 3, \\ O (p^{3n+3}) & {\mathrm {if\ }} \alpha = 4, \end {array}\right .\] we then infer from (33) and (57) the desired lemma as follows: Lemma 15 If$$p {{\vert }} {{{\bf{{m}}}}}$$, we have  \[Q ( {{{\bf{{m}}}}}, p^{\alpha }) = O (p^{n \alpha -n+3})\]when$$\alpha = 3$$and$$4$$. Note the result is trivial if $$p <p_0$$. Also, because of the comments at the end of the previous section, it is natural to extend the definition of $$r ( {{{\bf{{m}}}}}, p)$$ by assigning to it the value $$1$$ when $$p {{\vert }} {{{\bf{{m}}}}}$$, or, in other words, when $$ {{{\bf{{m}}}}}$$ in $${\Bbb {F}}_p^{n}$$ belongs to the zero-dimensional set consisting of the origin. With this understanding, we see that the estimate in Lemma 13 is also valid for $$r=1$$, although the same cannot be said about Lemma 12. 10. Sums of $$Q ( {{{\bf{{m}}}}}, l_3)$$: first type Lemmas 12– 15, which have no counterpart in previous work on cubic forms, have a strength that is crucial to the success of our present researches. Nevertheless, they are not powerful enough in themselves to secure our result and it is necessary to complement them by estimates for certain sums of $$|{Q ( {{{\bf{{m}}}}}, l_3)}|$$ taken over various sets of $$ {{{\bf{{m}}}}}$$. To do this we need to extend certain notations that were introduced in Ha and Ib. Throughout the symbol $$l_r$$ denotes a positive integer whose prime divisors occur with exponent at least $$r$$, while $$k_r$$ is to be a positive integer (possibly $$1$$) whose prime divisors occur with exponent exactly equal to $$r$$ so that it is the $$r$$th power of a square-free number that will be designated $$\kappa _r$$. Thus any positive integer $$k$$ can be expressed (uniquely) as   \[k_1 k_2 \cdots k_r \cdots = k_1 k_2 \cdots k_r l_{r+1} = \kappa _1 \kappa _2^2 \cdots \kappa _r^r l_{r+1},\] (58) where the individual factors in each product are all coprime to each other; in this it will sometimes be convenient to write   \[k^{ * } = k_1 k_2.\] (59) Also, in what follows in this and the next section, we confine ourselves to the case where $$k$$ is a number of type $$l_3$$, in which circumstances we can write $$k$$ uniquely as   \[q_1^2 q_2 {\mathrm {\ where\ }} q_2 {\mathrm {\ is\ square-free\ and\ }} q_2 {{\vert }} q_1.\] (60) We then first cite an important estimate due to Heath-Brown (Ha; also enunciated in Ib). This is that the sum   \[\Omega ^{(1)} (y; l_3) = \sum _{{||{ {{{\bf{{m}}}}} }|| \leq y}} |{Q( {{{\bf{{m}}}}}, l_3)}|\] is   \[O({l_3^{({1}/{2}) n +1 + \epsilon } (l_3^{({1}/{3}) n} +y^n)}).\] (61) This lemma is adequate in all but one of the situations in which sums associated with those of the type $$\Omega ^{(1)} (y; l_3)$$ occur in the analysis. But in the exceptional cases where $$l_3$$ is a number $$k_3$$ it is not sharp enough and needs to be supplanted by a more accurate result that depends on Lemma 12. The sum $$Q( {{{\bf{{m}}}}}, l)$$ being multiplicative, it is evident that we can restrict attention to the case where all prime factors of $$k_3$$ exceed $$p_0$$ because of the shape of the estimate to be stated in (62). Furthermore, since also the only vectors contributing to the sum are those for which $$F ( {{{\bf{{m}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {\kappa _3}}$$, in view of Lemma 10, we can assume in place both here and later the structure involving $$r$$ and $$ {{{\bf{{m}}}}}$$ that was initiated in the middle of §7 and that was also discussed in the final parts of § § 8 and 9. So let us consider, for a given set of values $$r(p)$$ lying between $$1$$ and $$n$$ inclusive that are assigned for $$p {{\vert }} k_3$$, the sub-sum of $$\Omega ^{(1)} (y; k_3)$$ due to a summation over those vectors $$ {{{\bf{{m}}}}}$$ for which $$r( {{{\bf{{m}}}}}, p) = r(p)$$ throughout. Then, by Lemmas 12 and 15, the summand in this is   \[ {O \left({{A^{\omega (\kappa _3)} \prod _{p {\, {\vert } \,} \kappa _3} p^{2n +3 - ({1}/{2}) r(p) +R(p)}}}\right)},\] where $$R(p) = 0$$ or $$\frac {1}{2}$$ according as $$r(p) >1$$ or $$r(p) = 1$$. Also, by the definition of $$\dim (r,p)$$ in § 8 and the comments at the end of § 9, the cardinality of the vectors $$ {{{\bf{{m}}}}}$$ in the sub-summation is   \[ {O \left({{A^{\omega (\kappa _3)} \left({\frac {y^n}{\kappa _3^n} +1}\right) \prod _{p {\, {\vert } \,} \kappa _3} p^{\dim (r,p)}}}\right)},\] whence the sub-sum is   \[\begin {array}{rl} & {O \left({{A^{\omega (\kappa _3)} \left({\frac {y^n}{\kappa _3^n} +1}\right) \prod _{p {\, {\vert } \,} \kappa _3} p^{2n +3 + \dim (r,p) - ({1}/{2}) r(p) +R(p)}}}\right)} \\ & \quad = {O \left({{A^{\omega (\kappa _3)} \left( {\frac {y^n}{\kappa _3^n} +1}\right) \kappa _3^{({5}/{2}) n +2}}}\right)} \end {array}\] by a short verification involving the inequalities in (56). Since there are $$n^{\omega (\kappa _3)}$$ possible sub-sums and they exhaust all $$ {{{\bf{{m}}}}}$$ for $$Q ( {{{\bf{{m}}}}}, p^3) \neq 0$$, we gain the bound   \[\Omega ^{(1)} (y; l_3) = O({\kappa _3^{({3}/{2}) n +2 + \epsilon } (\kappa _3^{n} +y^{n})}) = O({k_3^{({1}/{2}) n + {2}/{3} + \epsilon } (k_3^{({1}/{3}) n} +y^{n})})\] (62) that replaces (61) when $$l_3 = k_3$$. Applying Lemma 5 to (61) and (62), we conclude our discussion of the first type of average by enunciating the following: Lemma 16 Suppose$$X,Y,$$and$$N$$are as in Lemma 5 and that$$l_3 <X^A$$. Then the sum  \[\Delta (Y, l_3) = \Delta _n (Y, l_3) = \sum _{{ {{{\bf{{m}}}}} \neq 0}} ||{ {{{\bf{{m}}}}} }||^{\epsilon } |{Q ( {{{\bf{{m}}}}}, l_3)}| J ( {{{\bf{{m}}}}}, Y)\] (63)is equal to  \[ {O \left({{\frac { {{\min}^{{\frac {1}{2} n - 1}}} (X, Y) X^{\epsilon } l_3^{({5}/{6}) n +1}}{X^{({1}/{2}) n - 1}}}}\right)} + {O \left({{\frac {Y^{({1}/{2}) n - 1} X^{\epsilon } N^{({1}/{2}) n +1} l_3^{({1}/{2}) n +1}}{X^{n}}}}\right)}.\] (64)Moreover, if$$l_3 = k_3,$$then the exponent of$$l_3$$in each term can be reduced by$$\frac {1}{3}$$. The sums over $$ {{{\bf{{m}}}}}$$ used in the estimation of $$\Upsilon _2 (X)$$ in (14) are of the type $$\Delta (Y, l_3)$$. But some of those needed in the estimation of $$\Upsilon _3 (X)$$ spring from the sums of the type   \[\Omega ^{(3)} (y; l_3) = \sum _{{\substack {0 <||{ {{{\bf{{m}}}}} }|| \leq y \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0}}} |{Q( {{{\bf{{m}}}}}, l_3)}|,\] (65) which, being much harder to estimate than $$\Omega ^{(1)} (y; l_3)$$, are treated in a separate section. 11. Sums of $$Q ( {{{\bf{{m}}}}}, l_3)$$: second type We begin by establishing an auxiliary estimate concerning the singular locus $$ {{\cal {{{S}}}}} ^{ {\dagger } }$$ of the dual $$ {{\cal {{{V}}}}_{{\mathrm {dual}}}}$$, which is known to have dimension $$n-2$$ in the affine sense. (All we need is that the dimension should not exceed $$n-2$$, the fact of which is clear from the absolute irreducibility of $$F( {{{\bf{{u}}}}})$$.) The subject of this is the number $$N (y; {{{\bf{{a}}}}}, k)$$ of integral elements $$ {{{\bf{{m}}}}}$$ of the type described by the conditions   \[||{ {{{\bf{{m}}}}} }|| \leq y, \quad {{{\bf{{m}}}}} \equiv {{{\bf{{a}}}}}, {\hbox{mod}\, {k}},\] (66) that satisfy the supplementary requirement that   \[{\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0.\] As a prelude to its assessment we require two estimates, the first of which is the relatively familiar lemma: Lemma 17 Let us temporarily allow$$ {{{\bf{{m}}}}}$$and$$ {{{\bf{{b}}}}}$$to contain$$s$$instead of$$n$$components and then let  \[\Theta ( {{\cal {{{Z}}}}} ^{ {\dagger } }) = \Theta ( {{\cal {{{Z}}}}} ^{ {\dagger } }; y; {{{\bf{{b}}}}}, k)\]be the number of integer points$$ {{{\bf{{m}}}}}$$of type (66)$$($$with$$ {{{\bf{{b}}}}}$$replacing$$ {{{\bf{{a}}}}})$$on an affine variety$$ {{\cal {{{Z}}}}} ^{ {\dagger } }$$defined over$${\Bbb {Q}}$$that lies in an ambient space of$$s$$dimensions. Then, if$$ {{\cal {{{Z}}}}} ^{ {\dagger } }$$be irreducible over$${\Bbb {Q}}$$and have dimension$$d,$$we have  \[\Theta ( {{\cal {{{Z}}}}} ^{ {\dagger } }) = {O \left({{\left( {\frac {y}{k} +1}\right)^d}}\right)}.\] The condition stated of irreducibility is clearly immaterial but we only use the lemma in the case when $$ {{\cal {{{Z}}}}} ^{ {\dagger } }$$ is irreducible. The second estimate is given by an important theorem by Cohen [1]. Lemma 18 According to the notation of the previous lemma, let now$$ {{\cal {{{Z}}}}} ^{ {\dagger } }$$be an affine conical hypersurface whose defining form$$\phi ( {{u}_{1}, \dotsc , {u}_{{s}}})$$is non-linear and irreducible over$${\Bbb {Q}}$$. Then  \[\Theta ( {{\cal {{{Z}}}}} ^{ {\dagger } }) = {O \left({{\left({\frac {y}{k} +1}\right)^{s - {3}/{2}} \log (y k +2)}}\right)}.\] From these lemmata, we can obtain what we seek in the form of the following lemma: Lemma 19 We have  \[N (y; {{{\bf{{a}}}}}, k) = {O \left({{\left({\frac {y}{k} +1}\right)^{n - {5}/{2}} \log (y k +2)}}\right)}.\] The affine cone $$ {{\cal {{{S}}}}} ^{ {\dagger } }$$ given by $${\mathrm {grad}} F( {{{\bf{{u}}}}}) = 0$$ is an irredundant union of affine cones $$ {{\cal {{{W}}}}} ^{ {\dagger } }_d$$ of dimension $$d \leq n-2$$ that are irreducible over $${\Bbb {Q}}$$. By Lemma 17, the contribution to $$N (y; {{{\bf{{a}}}}}, k)$$ of those $$ {{\cal {{{W}}}}} ^{ {\dagger } }_d$$ with $$d <n - 2$$ is   \[ {O \left({{\left( {\frac {y}{k} +1}\right)^{n - 3}}}\right)}.\] (67) However, no variety $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$ is a linear space because Lemma 31 of Ic states that $$ {{\cal {{{V}}}}_{{\mathrm {dual}}}}$$ does not contain a (projective) linear space of dimension $$n-3$$ when $$n \geq 5$$. Thus, by a familiar process to be briefly explained below, each $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$ is bi-rationally isomorphic to a cone that corresponds to an irreducible hypersurface of degree exceeding $$1$$. Indeed, let $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$ have a generic point $$( {{\xi }_{1}, \dotsc , {\xi }_{{n}}})$$, where, for example, $$ {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}}$$ are algebraically independent. Then, by the theory of the simplicity of algebraic extensions, let   \[\eta = a \xi _{n-1} +b \xi _{n}\] for suitable positive integers $$a$$ and $$b$$ so that $${\Bbb {Q}} ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}}, \xi _{n-1}, \xi _{n}) = {\Bbb {Q}} ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}}, \eta )$$, whence $$ {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}}, \eta$$ is a generic zero of an irreducible non-linear form $$\phi ( {{y}_{1}, \dotsc , {y}_{{n-1}}})$$ defined over $${\Bbb {Q}}$$ because not both $$\xi _{n-1}$$ and $$\xi _{n}$$ are of degree $$1$$ over $${\Bbb {Q}} ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}})$$ (that $$\phi ( {{y}_{1}, \dotsc , {y}_{{n-1}}})$$ is a form is clear from the fact that $$\lambda ( {{\xi }_{1}, \dotsc , {\xi }_{{n}}})$$ is also a generic point of $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$). Hence each point $$ {{{\bf{{m}}}}}$$ of type (66) on $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$ gives rise to a zero $$ {{{\bf{{m}}}}} ^{\prime }$$ of $$\phi ( {{y}_{1}, \dotsc , {y}_{{n-1}}})$$ for which   \[ {{{\bf{{m}}}}} ^{\prime } \equiv {{{\bf{{b}}}}}, {\hbox{mod}\, {k}}, \quad ||{ {{{\bf{{m}}}}} ^{\prime }}|| \leq (a+b) y\] where $$ {{{\bf{{b}}}}} = ( {{a}_{1}, \dotsc , {a}_{{n-2}}}, a a_{n-1} +b a_n)$$. Then, taking those $$ {{{\bf{{m}}}}} ^{\prime }$$ that spring from exactly one point $$ {{{\bf{{m}}}}}$$ in this construction, we see that the contribution of all the points of the latter type to $$N (y; {{{\bf{{a}}}}}, k)$$ is   \[ {O \left({{\left( {\frac {y}{k} +1}\right)^{n - {5}/{2}} \log (y k +2)}}\right)}\] (68) by Lemma 18. On the other hand, the inverse equations of the form:   \[\xi _{n-1} = \frac {h_1 ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}}, \eta )}{h_2 ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}})}, \ \xi _{n} = \frac {h_3 ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}}, \eta )}{h_4 ( {{\xi }_{1}, \dotsc , {\xi }_{{n-2}}})}\] certainly show that the $$ {{{\bf{{m}}}}}$$ of type (66) on $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$ that do not match a unique $$ {{{\bf{{m}}}}} ^{\prime }$$ meet an additional criterion of the type $$h_i ( {{m}_{1}, \dotsc , {m}_{{n-2}}}) = 0$$ ($$i = 2$$ or $$4$$). Since the resulting restriction of $$ {{\cal {{{W}}}}} ^{ {\dagger } }_{n-2}$$ is a variety of dimension $$n-3$$, the donation to $$N (y; {{{\bf{{a}}}}}, k)$$ of the remaining points $$ {{{\bf{{m}}}}}$$ is subject to another bound of type (67). Combining the bounds (67) with (68), we complete the proof of the lemma. The sum $$\Omega ^{(3)} (y; l_3)$$ defined by (65) is analogous to its namesake in (231) of Ic but, in spite of the non-singularity of $$f( {{{\bf{{x}}}}})$$ now assumed, the estimation is more onerous than before because of the demands placed on it in our overall method. Indeed, we now must yoke the procedure in §31 of Ic to the results of § § 8 and 9 in a complicated way, using in particular the function $$N (q; {{{\bf{{l}}}}})$$ that is the number of incongruent solutions, $$ {\hbox{mod}\, {q}}$$, of the congruence $$ {{{\bf{{M}}}}} ( {{{\bf{{l}}}}}) {{{\bf{{u}}}}} \equiv 0$$, $$ {\hbox{mod}\, {q}}$$, and that is subject to the relation   \[\sum _{{\substack {0 \leq {{{\bf{{l}}}}} <p \\ f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {p}} }}} N^{{1}/{2}} (p; {{{\bf{{l}}}}}) = O({p^{n-1}})\] (69) through Lemma 1 and the argument of Lemma 37 of Ic. First, by remarks similar to those in the treatment of $$\Omega ^{(1)} (y, k_3)$$ in § 10, we legitimately assume that in the factorization of $$q = l_3$$ as $$k_3 k_4 k_5 k_6 l_7$$ the prime divisors of $$k_4$$ and $$k_6$$ exceed $$p_0$$. Then to proceed we must delineate an intermediate sum that is to be estimated by a variant of a method used to deal with $$\Omega _0^{(3)} (y, l_3) = \Omega _0^{(3)} (y, q)$$ in Ic. This is to be the subject of a modulus $$q^{\prime }$$ of type $$l_3$$ that is expressed in terms of the factorization of $$q$$ as being   \[k_3 k_5 k_6^{\prime } l_7 = q_1^{\prime 2} q_2^{\prime } \quad (q_2^{\prime } {{\vert }} q_1^{\prime }; q_2^{\prime } {\mathrm {\ square-free}}),\] (70) where $$\kappa _6^{\prime } {{\vert }} \kappa _6$$ and where therefore $$q^{\prime } {{\vert }} q$$. A quasi-variety$$ {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }$$, $$ {\hbox{mod}\, {\kappa _6^{\prime }}}$$, having points with $$n$$ components is next to be introduced and consists of a complete set of incongruent elements, $$ {\hbox{mod}\, {\kappa _6^{\prime }}}$$, that are simultaneously congruent, $$ {\hbox{mod}\, {p}}$$, for all $$p {{\vert }} \kappa _6^{\prime }$$ to points belonging to affine conical varieties, $$ {\hbox{mod}\, {p}}$$, of dimension not exceeding $$4$$ and having a bounded number of components. The precise definition of $$ {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }$$ is immaterial for the moment but will be seen to be associated with the dimension of points $$ {{{\bf{{m}}}}}$$ for which $$r ( {{{\bf{{m}}}}},p)$$ has a given value. Then, to be used in conjunction with Lemmata 13– 15, the intermediate sums are of the type   \[\Omega ^{(4)} (y; {{{\bf{{a}}}}}, c; q^{\prime }) = \sum _{ {{{\bf{{b}}}}}_1 \in {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }} {\sum _{\substack {||{ {{{\bf{{m}}}}} }|| \leq y \\ {{{\bf{{m}}}}} \equiv {{{\bf{{a}}}}}, {\hbox{mod}\, {c}} \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0, {{{\bf{{m}}}}} \equiv {{{\bf{{b}}}}}_1, {\hbox{mod}\, {\kappa _6^{\prime }}} }}} |{Q ( {{{\bf{{m}}}}}, q^{\prime })}| \quad ( {({c}, {q^{\prime }})} = 1).\] (71) We follow closely the treatment of $$\Omega _0^{(3)} (y, l_3)$$ in §31 of Ic but modify it to take account of the fact that $$ {{\cal {{{V}}}}}$$ is now non-singular and that $$ {{\cal {{{W}}}}} ^{ {\dagger } }$$ in Ic is superseded by the affine $$ {{\cal {{{S}}}}} ^{\dagger }$$ corresponding to the singular locus of $$ {{\cal {{{V}}}}_{{\mathrm {dual}}}}$$. Accordingly, as in (232) of Ic, we have   \[\Omega ^{(4)} (y; {{{\bf{{a}}}}}, c; q^{\prime }) = {O \left({{q^{\prime n+1} q_2^{\prime ({1}/{2}) n +1} \sum _{ {{{\bf{{b}}}}}_1 \in {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }} {\sum _{\substack {0 <{{{\bf{{l}}}}}, h \leq q_1^{\prime } \\ {({h}, {q_1^{\prime }})} = 1; f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {q_1^{\prime }}} }}} N^{{1}/{2}} (q_2^{\prime }; {{{\bf{{l}}}}}) {\sum _{\substack {||{ {{{\bf{{m}}}}} }|| \leq y; {{{\bf{{m}}}}} \in {{\cal {{{S}}}}} ^{ {\dagger } } \\ {{{\bf{{m}}}}} \equiv h {\mathrm {grad}} f( {{{\bf{{l}}}}}), {\hbox{mod}\, {q_1^{\prime }}} \\ {{{\bf{{m}}}}} \equiv {{{\bf{{a}}}}}, {\hbox{mod}\, {c}}; {{{\bf{{m}}}}} \equiv {{{\bf{{b}}}}}_1, {\hbox{mod}\, {\kappa _6^{\prime }}} }}} 1}}\right)},\] (72) the development of which depends on an examination of those $$ {{{\bf{{l}}}}}$$, $$h$$ for which $$ {{{\bf{{m}}}}}$$ in the innermost sum lie on $$ {{\cal {{{S}}}}} ^{ {\dagger } }$$. Much as in Ic, letting   \[q_3^{\prime } = {\prod _{p {\, {\vert } \,} q_1^{\prime }, p \nmid q_2^{\prime }, p \nmid \kappa _6^{\prime }}} p\] that is relatively prime to $$\kappa _6^{\prime }$$, we define $$ {{\cal {{{S}}}}}_{q_3^{\prime }}^{ {\dagger } }$$ to be the quasi-reduction of $$ {{\cal {{{S}}}}} ^{\dagger }$$, $$ {\hbox{mod}\, {q_3^{\prime }}}$$, namely, a complete set of incongruent $$ {{{\bf{{b}}}}}_2$$, $$ {\hbox{mod}\, {q_3^{\prime }}}$$, satisfying the condition $${\mathrm {grad}} F( {{{\bf{{b}}}}}_2) \equiv 0$$, $$ {\hbox{mod}\, {q_3^{\prime }}}$$, whence we form the aggregate $$ {{\cal {{{P}}}}}_{q_3^{\prime } \kappa _6^{\prime }}^{ {\dagger } }$$ consisting of a full set of incongruent vectors $$ {{{\bf{{b}}}}}$$, $$ {\hbox{mod}\, {q_3^{\prime } \kappa _6^{\prime }}}$$, with the property that $$ {{{\bf{{b}}}}} \equiv {{{\bf{{b}}}}}_1$$, $$ {\hbox{mod}\, {\kappa _6^{\prime }}}$$, and $$ {{{\bf{{b}}}}} \equiv {{{\bf{{b}}}}}_2$$, $$ {\hbox{mod}\, {q_3^{\prime }}}$$, for $$ {{{\bf{{b}}}}}_1 \in {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }$$ and $$ {{{\bf{{b}}}}}_2 \in {{\cal {{{S}}}}}_{q_3^{\prime }}^{ {\dagger } }$$. Then, since the conditions of summation imply that $$h {\mathrm {grad}} f( {{{\bf{{l}}}}}) \equiv {{{\bf{{b}}}}}$$, $$ {\hbox{mod}\, {q_3^{\prime } \kappa _6^{\prime }}}$$, for some $$ {{{\bf{{b}}}}} \in {{\cal {{{P}}}}}_{q_3^{\prime } \kappa _6^{\prime }}^{ {\dagger } }$$,   \[\begin {array}{rl} \Omega ^{(4)} (y; {{{\bf{{a}}}}}, c; q^{\prime }) &= {O \left({{q^{\prime n+1} q_2^{\prime ({1}/{2}) n +1} \sum _{ {{{\bf{{b}}}}} \in {{\cal {{{P}}}}}_{q_3^{\prime } \kappa _6^{\prime }}^{ {\dagger } }} {\sum _{\substack {0 <{{{\bf{{l}}}}}, h \leq q_1^{\prime } \\ {({h}, {q_1^{\prime }})} = 1;\ f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {q_1^{\prime }}} \\ h {\mathrm {grad}} f( {{{\bf{{l}}}}}) \equiv {{{\bf{{b}}}}}, {\hbox{mod}\, {q_3^{\prime } \kappa _6^{\prime }}} }}} N^{{1}/{2}} (q_2^{\prime }; {{{\bf{{l}}}}}) {\sum _{\substack {||{ {{{\bf{{m}}}}} }|| \leq y; {{{\bf{{m}}}}} \in {{\cal {{{S}}}}} ^{ {\dagger } } \\ {{{\bf{{m}}}}} \equiv h {\mathrm {grad}} f( {{{\bf{{l}}}}}), {\hbox{mod}\, {q_1^{\prime }}} \\ {{{\bf{{m}}}}} \equiv {{{\bf{{a}}}}}, {\hbox{mod}\, {c}} }}} 1}}\right)} \\ &= {O \left({{q^{\prime n+1} q_2^{\prime \frac {1}{2} n +1} \sum _{ {{{\bf{{b}}}}} \in {{\cal {{{P}}}}}_{q_3^{\prime } \kappa _6^{\prime }}^{ {\dagger } }} {\sum _{\substack {0 <{{{\bf{{l}}}}}, h \leq q_1^{\prime } \\ {({h}, {q_1^{\prime }})} = 1;\ f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {q_1^{\prime }}} \\ h {\mathrm {grad}} f( {{{\bf{{l}}}}}) \equiv {{{\bf{{b}}}}}, {\hbox{mod}\, {q_3^{\prime } \kappa _6^{\prime }}} }}} N^{{1}/{2}} (q_2^{\prime }; {{{\bf{{l}}}}}) N (y; {{{\bf{{R}}}}}, q_1^{\prime } c)}}\right)}, \end {array}\] where $$ {{{\bf{{R}}}}}$$ is a simultaneous solution of $$ {{{\bf{{m}}}}} \equiv h {\mathrm {grad}} f( {{{\bf{{l}}}}})$$, $$ {\hbox{mod}\, {q_1^{\prime }}}$$, and $$ {{{\bf{{m}}}}} \equiv {{{\bf{{a}}}}}$$, $$ {\hbox{mod}\, {c}}$$. Hence, by Lemma 19 and the assumed coprimality of $$c$$ and $$q_1^{\prime }$$,   \[\begin {array}{rl} \Omega ^{(4)} (y; {{{\bf{{a}}}}}, c; q^{\prime }) &= O \left(q^{\prime n+1} q_2^{\prime ({1}/{2}) n +1} \left( {\frac {y}{q_1^{\prime } c} +1}\right)^{n - {5}/{2}} \log (y q_1^{\prime } c+1)\right.\\ & \quad \left.\times \sum _{ {{{\bf{{b}}}}} \in {{\cal {{{P}}}}}_{q_3^{\prime } \kappa _6^{\prime }}^{ {\dagger } }} {\sum _{\substack {0 <{{{\bf{{l}}}}}, h \leq q_1^{\prime } \\ {({h}, {q_1^{\prime }})} = 1; f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {q_1^{\prime }}} \\ h {\mathrm {grad}} f( {{{\bf{{l}}}}}) \equiv {{{\bf{{b}}}}}, {\hbox{mod}\, {q_3^{\prime } \kappa _6^{\prime }}} }}} N^{{1}/{2}} (q_2^{\prime }; {{{\bf{{l}}}}})\right ), \end {array}\] (73) the double sum in which is a multiplicative function in $$q^{\prime }$$ in an obvious sense and is temporarily denoted by $$\sum _{q^{\prime }}$$ for convenience. To assess $$\sum _{p^{\alpha ^{\prime }}}$$ for $$\alpha ^{\prime } \geq 3$$ let us suppose first that $$\alpha ^{\prime } = 2 \beta$$ so that the respective roles of $$q^{\prime }$$, $$q_1^{\prime }$$, $$q_2^{\prime }$$, and $$q_3^{\prime } \kappa _6^{\prime }$$ within the sum are played by $$p^{\alpha ^{\prime }}$$, $$p^{\beta }$$, $$1$$, and $$p$$, the inner sum being $$1$$; note that the case $$\alpha ^{\prime } = 4$$ does not occur, although this point is not yet vital. Then, if $$ {{{\bf{{b}}}}} {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p}}$$, the number of incongruent solutions, $$ {\hbox{mod}\, {p}}$$, of the conditions   \[0 <{{{\bf{{l}}}}}_1, h_1 \leq p; \ {({h_1}, {p})} = 1; \ f( {{{\bf{{l}}}}}_1) \equiv 0, {\hbox{mod}\, {p}}; \ h_1 {\mathrm {grad}} f( {{{\bf{{l}}}}}_1) \equiv {{{\bf{{b}}}}}, {\hbox{mod}\, {p}},\] (74) is seen to be $$O({p})$$ for $$p >p_0$$ by earlier comments about the geometry of the non-singular $$ {{\cal {{{V}}}}}_p$$, each $$ {{{\bf{{l}}}}}_1$$ appearing being indivisible by $$p$$; this remains true trivially for $$p \leq p_0$$. Hence, by a now familiar argument involving the solutions of $$f( {{{\bf{{l}}}}}) \equiv 0$$, $$ {\hbox{mod}\, {p^{\alpha }}}$$, for which $$ {{{\bf{{l}}}}} \equiv {{{\bf{{l}}}}}_1$$, $$ {\hbox{mod}\, {p}}$$, the inner sum in (73) is $$O({p^{\beta -1} \cdot p^{(n-1)(\beta -1)}) \cdot p} = O({p^{n(\beta -1) +1}})$$. Now, if $$\alpha ^{\prime } \neq 6$$, the dimension of $$ {{\cal {{{P}}}}}_{p}^{ {\dagger } }$$ is $$n-2$$ for $$p >p_0$$ and therefore for all $$p$$ the contribution to $$\sum _{p^{\alpha ^{\prime }}}$$ due to all those $$ {{{\bf{{b}}}}}$$ in $$ {{\cal {{{P}}}}}_{p}^{ {\dagger } }$$ that are indivisible by $$p$$ is $$O({p^{n-2} \cdot p^{n(\beta -1) +1}}) = O({p^{n\beta -1}})$$; yet, if $$\alpha = 6$$, the dimension of $$ {{\cal {{{P}}}}}_{p}^{ {\dagger } }$$ does not exceed $$4$$ because of the nature of $$\kappa _6^{\prime }$$ and $$ {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }$$, whence in this instance the corresponding contribution to $$\sum _{p^{\alpha ^{\prime }}}$$ is $$O({p^{4} \cdot p^{n(\beta -1) +1}}) = O({p^{n\beta -3}})$$ when $$n = 8$$. On the other hand, if $$ {{{\bf{{b}}}}} \equiv 0$$, $$ {\hbox{mod}\, {p}}$$, then (74) for $$p >p_0$$ implies that $$ {{{\bf{{l}}}}}_1 \equiv 0$$, $$ {\hbox{mod}\, {p}}$$, by the non-singularity of $$ {{\cal {{{V}}}}}_p$$ and we deduce that the inner sum in (73) is then not more than $$p^{\beta }$$ times the number of incongruent solutions, $$ {\hbox{mod}\, {p^{\beta -1}}}$$, of $$f( {{{\bf{{l}}}}} ^{\prime }) \equiv 0$$, $$ {\hbox{mod}\, {p^{\beta -3}}}$$, this product being $$p^{2n + \beta } \nu (p^{\beta -3}) = O({p^{2n + \beta } \cdot p^{(n-1)(\beta -3)}}) = O({p^{n \beta - n +3}}) = O({p^{n \beta -3}})$$ by (32); this estimate for the remaining contribution to $$\sum _{p^{\alpha ^{\prime }}}$$ is yet true in those cases where $$p <p_0$$ because of the crude assessment $$O({p^{\beta } \nu (p^{\beta })}) = O({p^{n \beta }}) = O({p^{n \beta -3}})$$ for the inner sum in (73). Summing up the estimates so far obtained, we conclude that   \[\sum _{p^{\alpha ^{\prime }}} = \left\{ \begin {array}{ll}O({p^{n \beta -1}}) & {\mathrm {if\ }} \alpha {\mathrm {\ even\ and\ }} \alpha \neq 6, \\ O({p^{n \beta -3}}) & {\mathrm {if\ }} \alpha = 6, \end {array}\right .\] (75) when $$n = 8$$. In the other case where $$\alpha ^{\prime }$$ is odd and equals to $$2 \beta +1$$ there is no condition involving $$ {{{\bf{{b}}}}}$$ and clearly   \[\sum _{p^{\alpha ^{\prime }}} = O\left (p^{\beta } \cdot p^{(n-1)(\beta -1)} {\sum _{\substack {0 <{{{\bf{{l}}}}} \leq p \\ f( {{{\bf{{l}}}}}) \equiv 0, {\hbox{mod}\, {p}} }}} N^{{1}/{2}} ( {{{\bf{{l}}}}}, p)\right ) = O({p^{\beta } \cdot p^{(n-1)(\beta -1)} \cdot p^{n-1}}) = O({p^{n \beta }})\] (76) by (69). Therefore, by multiplicativity and by combining (75) and (76) with (73), we derive the following: Lemma 20 Let$$\Omega ^{(4)} (y; {{{\bf{{a}}}}}, c; q^{\prime })$$be defined as in (71) when$$q^{\prime }$$is a number of the type in (70). Then, letting  \[q_4^{\prime } = {\prod _{\substack {p {\, {\vert } \,} q_1^{\prime }; p \nmid q_2^{\prime } \\ p \nmid \kappa _6}}} p \, O\left (\prod _{p {\, {\vert } \,} \kappa _6^{\prime }} p\right )^3,\]we have  \[\Omega ^{(4)} (y; {{{\bf{{a}}}}}, c; q^{\prime }) = O\left ({\frac {A^{\omega (q^{\prime })} q_1^{\prime 2n+1} q_2^{\prime \frac {1}{2} n +1}}{q_4^{\prime }} \left( {\frac {y}{q_1^{\prime } c} +1}\right)^{n - \frac {5}{2}} \log (y q_1^{\prime } c +2)}\right )\]when$$n = 8$$. We return to $$\Omega ^{(3)} (y; q)$$ and consider the contribution to it due to the vectors $$ {{{\bf{{m}}}}}$$ in the summation that satisfy all the following restraints, $$r( {{{\bf{{m}}}}}, p_4)$$ has a given value $$r(p_4)$$ for each prime factor $$p_4$$ of $$\kappa _4$$, $$r( {{{\bf{{m}}}}}, p_6) \leq 4$$ for the prime factors $$p_6$$ of a divisor $$\kappa _6^{\prime }$$ of $$\kappa _6$$ and $$r( {{{\bf{{m}}}}}, p_6)$$ has a given value $$r(p_6) >4$$ for the prime factors $$p_6$$ of the divisors $$\kappa _6^{\prime \prime }$$ of $$\kappa _6$$ conjugate to $$\kappa _6^{\prime }$$; thus $$\kappa _6 = \kappa _6^{\prime } \kappa _6^{\prime \prime }$$ where $$k_6^{\prime } = (\kappa _6^{\prime })^6$$, $$k_6^{\prime \prime } = (\kappa _6^{\prime \prime })^6$$, and $$k_6 = k_6^{\prime } k_6^{\prime \prime }$$. Finally, let us write   \[q^{\prime } = q / k_4 k_6^{\prime \prime },\] (77) at which modulus Lemma 20 will later be directed. Since Lemma 6 shows that   \[Q ( {{{\bf{{m}}}}}, q) = \prod _{{p_4 {\, {\vert } \,} \kappa _4}} Q ( {{{\bf{{m}}}}}, p_4^4) \prod _{{p_6 {\, {\vert } \,} \kappa _6^{\prime \prime }}} Q ( {{{\bf{{m}}}}}, p_6^6) \cdot Q ( {{{\bf{{m}}}}}, q^{\prime }),\] the summand in the portion of $$\Omega ^{(3)} (y, q)$$ isolated above is   \[O\left ({A^{\omega (\kappa _4 \kappa _6^{\prime \prime })} \prod _{{p_4 {\, {\vert } \,} \kappa _4}} p_4^{3n+4 - r(p_4)} \prod _{{p_6 {\, {\vert } \,} \kappa _6^{\prime \prime }}} p_6^{5n+6 - 2r(p_6)} \cdot |{Q ( {{{\bf{{m}}}}}, q^{\prime })}| }\right )\] because of Lemmata 13 and 14. But, by (56), the entities $$ {{{\bf{{m}}}}}$$ governing the summation belong to   \[O\left (A^{\omega (\kappa _4 \kappa _6^{\prime \prime })} \prod _{{p_4 {\, {\vert } \,} \kappa _4}} p_4^{r(p_4)} \prod _{{p_6 {\, {\vert } \,} \kappa _6^{\prime \prime }}} p_6^{r(p_6)}\right )\] residue classes, $$ {\hbox{mod}\, {\kappa _4 \kappa _6^{\prime \prime }}}$$, whence, considering the effect of each such residue class and taking into account the fact the $$ {{{\bf{{m}}}}}$$ are within a quasi-variety $$ {{\cal {{{T}}}}}_{\kappa _6^{\prime }}^{ {\dagger } }$$, we gain in all the estimate   \[\begin {array}{rl} & O\left (A^{\omega (\kappa _4 \kappa _6^{\prime \prime })} \kappa _4^{3n+4} \prod _{{p_6 {\, {\vert } \,} \kappa _6^{\prime \prime }}} p_6^{5n+6 - r(p_6)} \max _{0 <{{{\bf{{a}}}}} \leq \kappa _4 \kappa _6^{\prime \prime }} \Omega ^{(4)} (y; {{{\bf{{a}}}}}, \kappa _4 \kappa _6^{\prime \prime }; q^{\prime })\right ) \\ & \quad = O\left (A^{\omega (\kappa _4 \kappa _6^{\prime \prime })} \kappa _4^{3n+4} \kappa _6^{\prime \prime 5n+1} \max _{0 <{{{\bf{{a}}}}} \leq \kappa _4 \kappa _6^{\prime \prime }} \Omega ^{(4)} (y; {{{\bf{{a}}}}}, \kappa _4 \kappa _6^{\prime \prime }; q^{\prime })\right ) \end {array}\] (78) for the sub-sum over $$ {{{\bf{{m}}}}}$$ since $$r(p_6) >4$$ when $$p_6 {{\vert }} \kappa _6^{\prime \prime }$$. Hence, fixing the divisors $$\kappa _6^{\prime }$$, $$\kappa _6^{\prime \prime }$$ of $$\kappa _6$$ but otherwise summing this estimate over all admissible values of $$r(p_4)$$ and those of $$r(p_6)$$ exceeding $$4$$, we find that (78) with a different value of $$A$$ supplies the total contribution to $$\Omega ^{(3)} (y;q)$$ corresponding to those values of $$\kappa _6^{\prime }$$, $$\kappa _6^{\prime \prime }$$. This, by Lemma 20 with $$q^{\prime }$$ as in (77), is   \[\begin {array}{rl} & O\left (A^{\omega (q)} \kappa _4^{3n+4} \kappa _6^{\prime \prime 5n+1} \frac {q_1^{\prime 2n+1} q_2^{\prime ({1}/{2}) n+1}}{q_4^{\prime }} \left({\frac {y}{q_1^{\prime } \kappa _4 \kappa _6^{\prime \prime }} +1}\right)^{n - {5}/{2}} \log (yq +2)\right ) \\ & \quad = O\left (\frac {A^{\omega (q)} \kappa _4^{2n + {13}/{2}} \kappa _6^{\prime \prime 4n + {7}/{2}} q_1^{\prime n + {7}/{2}} q_2^{\prime \frac {1}{2} n+1} y^{n - {5}/{2}} \log (yq +2)}{q_4^{\prime }}\right ) \\ & \qquad +O\left (\frac {A^{\omega (q)} \kappa _4^{3n+4} \kappa _6^{\prime \prime 5n+1} q_1^{\prime 2n+1} q_2^{\prime ({1}/{2}) n+1} \log (yq +2)}{q_4^{\prime }}\right ). \end {array}\] (79) We now introduce the notation $$ {{\mathrm {{q}}}} = q^{\prime } / k_6^{\prime }$$, viz. $$ {{\mathrm {{q}}}} = q / k_4 k_6$$, assigning the obvious meanings to $$ {{\mathrm {{q}}}}_1$$, $$ {{\mathrm {{q}}}}_2$$ and letting   \[ {{\mathrm {{q}}}}_3 = {\prod _{p {\, {\vert } \,} {{\mathrm {{q}}}}_1; p \nmid {{\mathrm {{q}}}}_2}} p.\] Then, as $$q_1^{\prime } = \kappa _5^{\prime 3} {{\mathrm {{q}}}}_1$$, $$q_2^{\prime } = {{\mathrm {{q}}}}_2$$, and $$q_4^{\prime } = \kappa _6^{\prime 3} {{\mathrm {{q}}}}_3$$ by (77), the estimate (79) becomes   \[\begin {array}{rl} & O\left (\frac {A^{\omega (q)} \kappa _4^{2n+ {13}/{2}} \kappa _6^{\prime 3n+ {15}/{2}} \kappa _6^{\prime \prime 4n+ {7}/{2}} {{\mathrm {{q}}}}_1^{n+ {7}/{2}} {{\mathrm {{q}}}}_2^{({1}/{2}) n+1} y^{n - {5}/{2}} \log (yq +2)}{ {{\mathrm {{q}}}}_3}\right ) \\ & \quad {+ } O\left (\frac {A^{\omega (q)} \kappa _4^{3n+4} \kappa _6^{\prime 6n} \kappa _6^{\prime \prime 5n+1} {{\mathrm {{q}}}}_1^{2n+1} {{\mathrm {{q}}}}_2^{({1}/{2}) n+1} \log (yq +2)}{ {{\mathrm {{q}}}}_3}\right )\\ & \qquad = O\left (\frac {A^{\omega (q)} \kappa _4^{2n+ {13}/{2}} \kappa _6^{4n+ {7}/{2}} {{\mathrm {{q}}}}_1^{n+ {7}/{2}} {{\mathrm {{q}}}}_2^{({1}/{2}) n+1} y^{n - {5}/{2}} \log (yq +2)}{ {{\mathrm {{q}}}}_3}\right )\\ & \qquad \quad {+ } O\left (\frac {A^{\omega (q)} \kappa _4^{3n+4} \kappa _6^{6n} {{\mathrm {{q}}}}_1^{2n+1} {{\mathrm {{q}}}}_2^{({1}/{2}) n+1} \log (yq +2)}{ {{\mathrm {{q}}}}_3}\right ), \end {array}\] (80) whence, summing over all solutions of $$\kappa _6 = \kappa _6^{\prime } \kappa _6^{\prime \prime }$$ which are $$O({A^{\omega (\kappa _6)}})$$ in number, we deduce that $$\Omega ^{(3)} (y;q)$$ is subject to the bound (80) with yet a different value of $$A$$. Thus, finally, referring to Lemma 5, we infer the following result that it was the object of this section to reach. Lemma 21 Suppose$$X$$,$$Y$$, and$$N$$are as in the statement of Lemma 5 and that$$q <X^A$$. Then for$$n=8$$the sum  \[ {{\mathrm {P}}} (Y, q) = \sum _{{\substack { {{{\bf{{m}}}}} \neq 0 \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0}}} ||{ {{{\bf{{m}}}}} }||^{\epsilon } |{Q ( {{{\bf{{m}}}}}, q)}| J ( {{{\bf{{m}}}}}, Y)\]is equal to  \[\begin {array}{rl} & O\left (\frac {Y^{({1}/{2}) n -1} X^{\epsilon } N^{({1}/{2}) n - {3}/{2}} \kappa _4^{2n + {13}/{2}} \kappa _6^{4n + {7}/{2}} {{\mathrm {{q}}}}_1^{n + {7}/{2}} {{\mathrm {{q}}}}_2^{({1}/{2}) n +1}}{ {{\mathrm {{q}}}}_3 X^{n - {5}/{2}}}\right )\\ & \quad +O\left (\frac {\min ^{({1}/{2}) n -1} (X, Y) X^{\epsilon } \kappa _4^{3n+4} \kappa _6^{6n} {{\mathrm {{q}}}}_1^{2n+1} {{\mathrm {{q}}}}_2^{({1}/{2}) n +1}}{ {{\mathrm {{q}}}}_3 X^{({1}/{2}) n - 1}}\right )\end {array}\] (81)  \[\qquad = O\left (\frac {Y^{({1}/{2}) n -1} X^{\epsilon } N^{({1}/{2}) n - {3}/{2}}}{X^{n - {5}/{2}}} G_1(q)\right ) +O\left (\frac {\min ^{({1}/{2}) n -1} (X, Y) X^{\epsilon }}{X^{({1}/{2}) n - 1}} G_2(q)\right ) \quad {\mathrm {say}},\] (82)where$$q = k_4 k_6 {{\mathrm {{q}}}}$$in accordance with earlier notation. We need to determine the respective exponents $$\gamma _{r,1}$$ and $$\gamma _{r,2}$$ of the powers of $$\kappa _{r}$$ that enter into $$G_1(q)$$ and $$G_2(q)$$. Evidently   \[\gamma _{r,1} = \left\{ \begin {array}{ll}2n+ \frac {13}{2} & {\mathrm {if\ }} r=4, \\ 4n+ \frac {7}{2} & {\mathrm {if\ }} r=6, \end {array}\right . \quad \gamma _{r,2} = \left\{ \begin {array}{ll}3n+4 & {\mathrm {if\ }} r=4, \\ 6n & {\mathrm {if\ }} r=6, \end {array}\right .\] (83) but, otherwise, setting $$r = 2 \beta$$ or $$2 \beta +1$$ so that   \[\kappa _{r}^{\beta }\|{\mathrm {q}}_1, \kappa _{r} \nmid {\mathrm {q}}_2, \kappa _{r} \|{\mathrm {q}}_3, {\mathrm {\ or\ }} \kappa _{r}^{\beta }\|{\mathrm {q}}_1, \kappa _{r} \| {\mathrm {q}}_2, (\kappa _{r},{\mathrm {q}}_3)=1,\] we find after a short calculation that for $$r \neq 4, 6$$  \[\gamma _{r,1} = \left\{ \begin {array}{ll}\frac {1}{2} r (n+ \frac {7}{2}) -1 & {\mathrm {if\ }} r {\mathrm {\ even}}, \\ \frac {1}{2} r (n+ \frac {7}{2}) - \frac {3}{4} & {\mathrm {if\ }} r {\mathrm {\ odd}}, \end {array}\right . \quad \gamma _{r,2} = \left\{ \begin {array}{ll}\frac {1}{2} r (2n+1) -1 & {\mathrm {if\ }} r {\mathrm {\ even}}, \\ \frac {1}{2} r (2n+1) - \frac {1}{2} n + \frac {1}{2} & {\mathrm {if\ }} r {\mathrm {\ odd}}. \end {array}\right .\] (84) From a scrutiny of these indices, we infer a simplified version of the lemma that will be useful in some of the less delicate analysis that follows. Corollary (Lemma 21) Under the conditions laid out in Lemma 21 with$$q = k_3 l_4,$$we have  \[G_1 (k_3 l_4) = k_3^{({1}/{2}) n + {3}/{2}} l_4^{({2}/{3}) n + {2}/{3}} \quad {\mathrm {and}}\quad G_2 (k_3 l_4) = k_3^{({5}/{6}) n + {2}/{3}} l_4^{n + {1}/{2}}.\] In ending this section we should explain that a direct modification of our processes in Ic would have led to a similar result in which $$G_1(q)$$ would be replaced by something actually a bit better but in which $$G_2(q)$$ was somewhat worse. But the success of the later analysis is more dependent on the sharpness of the estimation of $$G_2(q)$$ rather than on that of $$G_1(q)$$ even though the latter cannot be allowed to be too blunt. 12. The sums $$S( {{{\bf{{m}}}}}, y)$$ and Hypothesis HW We look back at the sums $$Q( {{{\bf{{m}}}}}, k)$$ for square-free $$k = k_1$$, upon which we touched in § 6, and begin by discussing an estimate for them before considering how sums of the type   \[S( {{{\bf{{m}}}}}, y) = \sum _{{\substack {k_1 \leq y \\ {({k_1}, {F( {{{\bf{{m}}}}})})} = 1}}} \frac {Q( {{{\bf{{m}}}}}, k_1)}{k_1^{{9}/{2}}} \quad (F( {{{\bf{{m}}}}}) \neq 0)\] (85) can be bounded for $$n=8$$ through the assumption of the Hypothesis HW. We follow for the most part the method of §6 of Ia and therefore need only to supply an attenuated explanation, it being assumed throughout that $$F( {{{\bf{{m}}}}}) \neq 0$$. Of the entities $$ {{\cal {{{V}}}}}_p$$ and $$ {{\cal {{{V}}}}}_p ( {{{\bf{{m}}}}})$$ that are defined in § 3, the first is a non-singular projective hypersurface of dimension $$n-2$$ and the second is a non-singular projective hypersurface of dimension $$n-3$$ embedded in projective space of dimension $$n-1$$ when $$p \nmid F( {{{\bf{{m}}}}})$$. Next, if $$\rho (p)$$ and $$\rho ( {{{\bf{{m}}}}}, p)$$ denote, respectively, the number of (projective) points with coordinates in $${\Bbb {F}}_p$$ that lie on $$ {{\cal {{{V}}}}}_p$$ and $$ {{\cal {{{V}}}}}_p ( {{{\bf{{m}}}}})$$, then we have   \[\nu (p) = (p-1) \rho (p) +1, \quad \nu ( {{{\bf{{m}}}}}, p) = (p-1) \rho ( {{{\bf{{m}}}}}, p) +1,\] when we use the notation introduced at the beginning of § 6. Also, by Deligne's theory [3], we have   \[\rho (p) = \frac {p^{n-1} -1}{p-1} +O({p^{({1}/{2}) (n-2)}})\] and   \[\rho ( {{{\bf{{m}}}}}, p) = \frac {p^{n-2} -1}{p-1} - \sum _{{1 \leq j \leq B_n}} \lambda _{j,p} = \frac {p^{n-2} -1}{p-1} +O({p^{({1}/{2}) (n-3)}}),\] where $$|{\lambda _{j,p}}| = |{\lambda _{j,p, {{{\bf{{m}}}}} }}| = p^{({1}/{2}) (n-3)}$$ and   \[B_n = \frac {2}{3} ({2^{n-2} + (-1)^{n-1}})\] when $$p \nmid F( {{{\bf{{m}}}}})$$. Then, by way of Lemma 7, we obtain the estimate   \[\begin {array}{rl} Q( {{{\bf{{m}}}}},p) &= p (p \rho ( {{{\bf{{m}}}}}, p) - \rho (p) +1) \\ &= p \left( {\frac {p^{n-1} - p}{p-1} - \frac {p^{n-1} - 1}{p-1} +1}\right) +O({p^{({1}/{2}) (n+1)}}) +O({p^{({1}/{2}) n}}) \\ &= O({p^{({1}/{2}) (n+1)}}) \end {array}\] (86) that confirms Lemma 8 for the case $$F( {{{\bf{{m}}}}}) {{\not\equiv }} 0$$, $$ {\hbox{mod}\, {p}}$$; a variation in this process substantiates the lemma in other cases. We thus obtain at once the bound   \[\sum _{{\substack {k_1 \leq y \\ {({k_1}, {F( {{{\bf{{m}}}}})})} = 1}}} \,\frac { |{Q( {{{\bf{{m}}}}}, k_1)}| }{k_1^{({1}/{2}) (n+1)}} = O({||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{1 + \epsilon }}),\] which, however, for $$n=8$$ is insufficiently sharp for our purposes. We therefore intend to reduce the exponent of $$y$$ in this by $$\frac {1}{2}$$ by assuming the relevant form of Hypothesis HW for $$n=8$$ which we shall shortly enunciate. Rephrasing (86) in a more explicit form when $$n=8$$, we have the equation   \[Q_3 ( {{{\bf{{m}}}}}, p) = \frac {Q( {{{\bf{{m}}}}},p)}{p^2} = - \sum _{{1 \leq j \leq 42}} \lambda _{j,p} + {O \left({{p^2}}\right)},\] which we associate with the local$$L$$-function   \[L_p ( {{{\bf{{m}}}}}, s) = \prod _{1 \leq j \leq 42} \left({1- \frac {\lambda _{j,p}}{p^s}}\right)^{-1}\] for $$p \nmid F( {{{\bf{{m}}}}})$$ by comparing the equalities   \[1 + \frac {Q_3 ( {{{\bf{{m}}}}},p)}{p^s} = 1 - \frac {1}{p^s} {\sum _{1 \leq j \leq 42}} \lambda _{j,p} + {O \left({{\frac {1}{p^{\sigma -2}}}}\right)}\] and   \[\frac {1}{L_p ( {{{\bf{{m}}}}},s)} = 1 - \frac {1}{p^s} {\sum _{1 \leq j \leq 42}} \lambda _{j,p} + {O \left({{\frac {1}{p^{2 \sigma -5}}}}\right)}.\] Hence, bringing in the Hasse–Weil $$L$$-function   \[L^{ * } ( {{{\bf{{m}}}}},s) = \prod _{{p \nmid F( {{{\bf{{m}}}}})}} L_p ( {{{\bf{{m}}}}},s)\] and the function   \[\Psi ( {{{\bf{{m}}}}},s) = \sum _{{\substack {k_1 \\ {({F( {{{\bf{{m}}}}})}, {k_1})} = 1}}} \,\frac {Q_3 ( {{{\bf{{m}}}}},k_1)}{k_1^s}\] that are defined initially for $$\sigma >\frac {7}{2}$$, we deduce that   \[\Psi ( {{{\bf{{m}}}}},s) = \frac {\Theta ( {{{\bf{{m}}}}},s)}{L^{ * } ( {{{\bf{{m}}}}},s)},\] (87) in which   \[\Theta ( {{{\bf{{m}}}}},s) = \prod _{{p \nmid F( {{{\bf{{m}}}}})}} \left( {1 + {O \left({{\frac {1}{p^{\sigma -2}}}}\right)} }\right)\] is regular and bounded for $$\sigma >3$$. The stage has been set for the entrance of a modified Hasse–Weil $$L$$-function according to the tenets of Serre [15]. For each prime $$p$$ dividing $$F( {{{\bf{{m}}}}})$$ we take a certain factor   \[L_p ( {{{\bf{{m}}}}},s) = \prod _{j} \left({1 - \frac {\lambda _{j,p}}{p^s}}\right)^{-1}\] that is defined by Serre in such a way that $$|{\lambda _{j,p}}| = p^{({1}/{2}) a_j}$$ for some non-negative integer $$a_j$$ not exceeding $$5$$. Then the new $$L$$-function is   \[L( {{{\bf{{m}}}}},s) = L^{ * } ( {{{\bf{{m}}}}},s) \, \prod _{{p {\, {\vert } \,} F( {{{\bf{{m}}}}})}} L_p ( {{{\bf{{m}}}}},s) = L^{ * } ( {{{\bf{{m}}}}},s) \Lambda ( {{{\bf{{m}}}}},s) \quad {\mathrm {say}},\] (88) whose conductor $$B( {{{\bf{{m}}}}})$$ is regarded as being   \[\prod _{{p {\, {\vert } \,} F( {{{\bf{{m}}}}})}} p^{b_p}\] for bounded exponents $$b_p$$ and is therefore $${O({||{ {{{\bf{{m}}}}} }||^A})}$$. Associated with this, there is the function (defined initially for $$\sigma >\frac {7}{2}$$)   \[\xi ( {{{\bf{{m}}}}},s) = (2 \pi )^{-21\,s} \Gamma ^{21} (s-2) B^{({1}/{2}) s} ( {{{\bf{{m}}}}}) L ( {{{\bf{{m}}}}},s),\] about which in conformity with Serre's conjecture we enunciate the following: Hypothesis HW If $$F( {{{\bf{{m}}}}}) \neq 0$$, then $$\xi ( {{{\bf{{m}}}}},s)$$ is a meromorphic function of finite order that is regular everywhere save possibly for poles at $$s = \frac {7}{2}$$ and $$\frac {5}{2}$$, $$\xi ( {{{\bf{{m}}}}},s)$$ satisfies the functional equation   \[\xi ( {{{\bf{{m}}}}},s) = w( {{{\bf{{m}}}}}) \xi ( {{{\bf{{m}}}}}, 6-s),\] where $$w( {{{\bf{{m}}}}}) = \pm 1$$, $$\xi ( {{{\bf{{m}}}}},s) \neq 0$$ if $$\sigma \neq 3$$ (Riemann hypothesis). Since the Dirichlet's series generating the sum $$S( {{{\bf{{m}}}}},y)$$ in (85) is   \[\frac {\Theta ( {{{\bf{{m}}}}}, s + {s}/{2}) \Lambda ( {{{\bf{{m}}}}}, s + {s}/{2})}{L^{ * } ( {{{\bf{{m}}}}}, s + {s}/{2})}\] by (87) and (88), we then find on Hypothesis HW that   \[S( {{{\bf{{m}}}}}, y) = O({||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{{1}/{2} + \epsilon }})\] (89) by following the proof of Lemma 10 in Ia. In final preparation for the estimation of $$\Psi _2 (Y)$$ we shall need a bound for the sum $$s( {{{\bf{{m}}}}}, y)$$ derived from $$S( {{{\bf{{m}}}}}, y)$$ by deleting the condition $$ {({k_1}, {F( {{{\bf{{m}}}}})})} = 1$$ in the summation. This is furnished by the following: Lemma 22 For$$F( {{{\bf{{m}}}}}) \neq 0$$,  \[s( {{{\bf{{m}}}}}, y) = O({||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{{1}/{2} + \epsilon }}).\] Indeed, using (89) and Lemma 8, we have   \[\begin {array}{rl} s( {{{\bf{{m}}}}}, y) &= \sum _{{\substack {k_1 k_1^{\prime } \leq y \\ k_1 {\, {\vert } \,} F( {{{\bf{{m}}}}}), {({k_1^{\prime }}, {F( {{{\bf{{m}}}}})})} = 1}}} \frac {Q ( {{{\bf{{m}}}}}, k_1 k_1^{\prime })}{k_1^{{9}/{2}} k_1^{\prime {9}/{2}}} = {\sum _{\substack {k_1 \leq y \\ k_1 {\, {\vert } \,} F( {{{\bf{{m}}}}})}}} \frac {Q ( {{{\bf{{m}}}}}, k_1)}{k_1^{{9}/{2}}} {\sum _{\substack {k_1^{\prime } \leq y / k_1 \\ {({k_1^{\prime }}, {F( {{{\bf{{m}}}}})})} = 1}}} \frac {Q ( {{{\bf{{m}}}}}, k_1^{\prime })}{k_1^{\prime {9}/{2}}} \\ &= O\left (||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{{1}/{2} + \epsilon } \sum _{{k_1 {\, {\vert } \,} F( {{{\bf{{m}}}}})}} \frac { |{Q ( {{{\bf{{m}}}}}, k_1)}| }{k_1^5}\right )\\ &= O\left (||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{{1}/{2} + \epsilon } \sum _{{k_1 {\, {\vert } \,} F( {{{\bf{{m}}}}})}} A^{\omega (k_1)}\right ) \\ &= O({ ||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{{1}/{2} + \epsilon } (A+1)^{\omega (F( {{{\bf{{m}}}}}))}}) = O({||{ {{{\bf{{m}}}}} }||^{\epsilon } y^{{1}/{2} + \epsilon }}). \end {array}\] 13. Sums involving the numbers $$l_r$$ The following treatments of $$\Upsilon _2 (X)$$ and $$\Upsilon _3 (X)$$ through what we have so far obtained will incidentally bring out the appearance of sums in which numbers of the type $$l_r$$ or $$k_r$$ appear. All are readily estimated and, by way of examples, it suffices to cite the bounds   \[\sum _{{l_r \leq \xi }} 1 = O({\xi ^{{1}/{r}}}),\quad \sum _{l_r >\xi } \frac {1}{l_r^{r_1}} = O\left (\frac {1}{\xi ^{r_1 - r^{-1}}}\right ) \quad \left (r_1 >\dfrac {1}{r}\right ),\] the first of which is the subject of Lemma 21 in Ib. 14. Estimation of $$\Upsilon _2 (X)$$ On the immediate assumption that $$n=8$$ and $$Y \leq A_1 N$$ as in (13), equation (14) implies that the segment $$\Psi _2 (Y)$$ of $$\Upsilon _2 (X)$$ is the sum   \[\sum _{F( {{{\bf{{m}}}}}) \neq 0} {\sum _{({1}/{2}) Y <k \leq Y}} \frac {Q( {{{\bf{{m}}}}},k)}{k^8} I_k ( {{{\bf{{m}}}}})\] (90) containing an inner sum that in the notation of (58) is equal to   \[\begin {array}{rl} & \sum _{{({1}/{2}) Y <k_1 k_2 l_3 \leq Y}} \frac {Q( {{{\bf{{m}}}}},k_1) Q( {{{\bf{{m}}}}},k_2) Q( {{{\bf{{m}}}}},l_3)}{k_1^8 k_2^8 l_3^8} I_{k_1 k_2 l_3} ( {{{\bf{{m}}}}}) \\ &\quad = \sum _{{k_2 l_3 \leq Y}} \frac {Q( {{{\bf{{m}}}}}, k_2) Q( {{{\bf{{m}}}}}, l_3)}{k_2^8 l_3^8} {\sum _{({1}/{2}) Y / k_2 l_3 <k_1 \leq Y / k_2 l_3}} \frac {Q( {{{\bf{{m}}}}}, k_1)}{k_1^8} I_{k_1 k_2 l_3} ( {{{\bf{{m}}}}}) \\ &\quad = \sum _{{k_2 l_3 \leq Y}} \frac {Q( {{{\bf{{m}}}}}, k_2) Q( {{{\bf{{m}}}}}, l_3)}{k_2^8 l_3^8} \sum _{Y, k_2 l_3, {{{\bf{{m}}}}} } \quad {\mathrm {say}}. \end {array}\] (91) Next, if $$s( {{{\bf{{m}}}}}, y)$$ be defined as in § 12, the inner sum in (91) can be expressed as   \[\begin {array}{rl} \int _{Y / 2 k_2 l_3}^{Y / k_2 l_3} \frac {I_{y k_2 l_3} ( {{{\bf{{m}}}}})}{y^{{7}/{2}}} {d} s( {{{\bf{{m}}}}}, y) & = {}^{Y / k_2 l_3}_{Y / 2 k_2 l_3}{ \left [{{\frac {s( {{{\bf{{m}}}}}, y) I_{y k_2 l_3} ( {{{\bf{{m}}}}})}{y^{{7}/{2}}}}}\right ] } \\ & \quad - \int _{Y / 2 k_2 l_3}^{Y / k_2 l_3} s( {{{\bf{{m}}}}}, y) \left (\frac {-7 I_{y k_2 l_3} ( {{{\bf{{m}}}}})}{2 y^{{9}/{2}}} + \frac {1}{y^{{7}/{2}}} \frac{\partial}{\partial y} I_{y k_2 l_3} ( {{{\bf{{m}}}}})\right ) {d} y, \end {array}\] whence, bringing (31) and Lemma 22 into play when $$F( {{{\bf{{m}}}}}) \neq 0$$, we deduce that   \[\begin {array}{rl} \sum _{Y, k_2 l_3, {{{\bf{{m}}}}} } &= \left (\frac {J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon } k_2^3 l_3^3}{Y^3}\right ) + \left (J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon } \int _{Y / 2 k_2 l_3}^{Y / k_2 l_3} \frac { {d} y}{y^4}\right ) \\ & \quad +O\left (J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon } \int _{Y / 2 k_2 l_3}^{Y / k_2 l_3} \frac {k_2 l_3 {d} y}{k_2 l_3 y^4}\right )\\ & = O\left (\frac {J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon } k_2^3 l_3^3}{Y^3}\right ). \end {array}\] Consequently, the inner sum in (90) is   \[O\left (\frac {J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon }}{Y^3} \sum _{{k_2 l_3 \leq Y}} \frac { |{Q( {{{\bf{{m}}}}}, k_2)}| |{Q( {{{\bf{{m}}}}}, l_3)}| }{k_2^5 l_3^5}\right )\] and this, by Lemma 9 and the bound   \[\sum _{l \leq \xi } \frac { {({b}, {l})} }{l} = O({d(b) \log 2 \xi }) \quad (b \neq 0)\] equals   \[\begin {array}{rl} & O\left (\frac {J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon }}{Y^3} \sum _{{l_3 \leq Y}} \frac { |{Q( {{{\bf{{m}}}}}, l_3)}| }{l_3^5} {\sum _{k_2 \leq Y / l_3}} \frac {A^{\omega (k_2)} {({F^2 ( {{{\bf{{m}}}}})}, {k_2})} ^{{1}/{2}}}{k_2^{{1}/{2}}}\right ) \\ &\quad = O\left (\frac {J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon }}{Y^3} \sum _{{l_3 \leq Y}} \frac { |{Q( {{{\bf{{m}}}}}, l_3)}| }{l_3^5} {\sum _{k_1^{\prime } \leq (Y / l_3)^{{1}/{2}}}} \frac { {({F( {{{\bf{{m}}}}})}, {k_1^{\prime }})} }{k_1^{\prime }}\right ) \\ &\quad = O\left (\frac {J( {{{\bf{{m}}}}}, Y) X^{\epsilon } ||{ {{{\bf{{m}}}}} }||^{\epsilon }}{Y^3} \sum _{{l_3 \leq Y}} \frac { |{Q( {{{\bf{{m}}}}}, l_3)}| }{l_3^5}\right ). \end {array}\] Therefore, in completion of the first stage of estimation, we get   \[\begin {array}{rl} \Psi _2 (Y) &= O\left (\frac {X^{\epsilon }}{Y^3} \sum _{l_3 \leq Y} \frac {1}{l_3^5} \sum _{ {{{\bf{{m}}}}} \neq 0} ||{ {{{\bf{{m}}}}} }||^{\epsilon } J( {{{\bf{{m}}}}}, Y) |{Q( {{{\bf{{m}}}}}, l_3)}| \right )\\ &= O\left (\frac {X^{\epsilon }}{Y^3} \sum _{{l_3 \leq Y}} \frac {\Delta _8 (Y, l_3)}{l_3^5}\right ) = O\left (\frac {X^{\epsilon } \Box (Y)}{Y^3}\right ) \quad {\mathrm {say}}, \end {array}\] (92) by (90) and (63). The estimation of $$\Box (Y)$$ is harder than might be expected at first sight because the first part of Lemma 16 is not quite sharp enough for our purpose. It is this feature that made it necessary to establish the second part of the lemma and hence the material of § 7, upon which it depends. Then, to combine the forces of both parts of the lemma, we write $$l_3$$ in the summation as $$k_3 l_4$$ and consider separately the two cases: $$l_4 \leq Y^{\delta }$$, $$l_4 >Y^{\delta }$$, where $$\delta$$ is a suitable small positive number. Since trivially $$|{Q( {{{\bf{{m}}}}}, l_4)}| \leq l_4^9$$ and therefore   \[\Delta _8 (Y, k_3 l_4) \leq l_4^9 \Delta _8 (Y, k_3),\] we infer by the second part of Lemma 16 that the impact on $$\Box (Y)$$ due to the first case is   \[\begin {array}{rl} & O\left (\sum _{l_4 \leq Y^{\delta }} \frac {l_4^9}{l_4^5} \left\{ \sum _{k_3 \leq Y} \left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon } k_3^{{22}/{3}}}{X^3 k_3^5} + \frac {Y^3 X^{\epsilon } N^5 k_3^{{14}/{3}}}{X^8 k_3^5}\right )\right\} \right ) \\ &\quad = O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon } Y^{5 \delta }}{X^3} \sum _{{k_3 \leq Y}} k_3^{{7}/{3}}\right ) +O\left (\frac {Y^{3+5 \delta } X^{\epsilon } N^5}{X^8} \sum _{{k_3 \leq Y}} \frac {1}{k_3^{{1}/{3}}}\right ) \\ &\quad = O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon } Y^{{8}/{3} +5 \delta }}{X^3}\right ) +O\left (\frac {Y^{3+5 \delta } X^{\epsilon } N^5}{X^8}\right ). \end {array}\] (93) On the other hand, by the first part of Lemma 16, the contribution corresponding to the second case is   \[O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon }}{X^3} \sum _{{\substack {k_3 l_4 \leq Y \\ l_4 >Y^{\delta }}}} (k_3 l_4)^{{8}/{3}}\right ) +O\left (\frac {Y^3 X^{\epsilon } N^5}{X^8} \sum _{{\substack {k_3 l_4 \leq Y \\ l_4 >Y^{\delta }}}} 1\right ),\] in which expression the first sum does not exceed   \[Y^{{8}/{3}} \sum _{l_4 >Y^{\delta }} {\sum _{k_3 \leq Y / l_4}} 1 = O\left (Y^3 \sum _{l_4 >Y^{\delta }} \frac {1}{l_4^{{1}/{3}}}\right ) = O\left (Y^{3 - {\delta }/{12}}\right )\] and the second is $$O({Y^{{1}/{3} - {\delta }/{12}}})$$. Hence, if   \[\frac {8}{3} +5 \delta = 3 - \frac {\delta }{12}\] so that $$\delta = \frac {4}{61}$$, the second contribution is formally equal to (93) and we conclude that   \[\Psi _2 (Y) = O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon }}{X^3 Y^{{\delta }/{12}}}\right ) +O\left (\frac {Y^{{20}/{61}} X^{\epsilon } N^5}{X^8}\right )\] for $$Y \leq A N$$ as in (15). Consequently, the estimate   \[\begin {array}{rl} \Upsilon _2 (X) &= O\left (\frac {X^{\epsilon }}{X^{{\delta }/{12}}}\right ) +O\left (\frac {X^{\epsilon }}{X^{{\delta }/{12}}}\right ) +O\left (\frac {X^{\epsilon } N^{{325}/{61}}}{X^8}\right )\\ &= O\left (\frac {X^{\epsilon }}{X^{{\delta }/{12}}}\right ) +O\left (\frac {X^{\epsilon }}{X^{{1}/{122}}}\right ) \\ &= {o (1)} \quad (X \to \infty ) \end {array}\] (94) is deduced in the usual manner. 15. Estimation of $$\Upsilon _3 (X)$$ By (14) and (31) the segment $$\Psi _3 (Y)$$ of $$\Upsilon _3 (X)$$ is   \[O\left (\frac {1}{Y^n} {\sum _{\substack { {{{\bf{{m}}}}} \neq 0 \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0}}} J( {{{\bf{{m}}}}}, Y) \sum _{{\frac {1}{2} Y <k \leq Y}} |{Q( {{{\bf{{m}}}}}, k)}| \right ) = O\left (\frac {1}{Y^n} {\sum _{\substack { {{{\bf{{m}}}}} \neq 0 \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0}}} J( {{{\bf{{m}}}}}, Y) \sum _{{k \leq Y}} |{Q( {{{\bf{{m}}}}}, k)}| \right ),\] in which through the notation (59) and Lemmata 8 and 9 the inner sum is   \[\begin {array}{rl} \sum _{{k^{ * } l_3 \leq Y}} |{Q( {{{\bf{{m}}}}}, k^{ * })}| |{Q( {{{\bf{{m}}}}}, l_3)}| &= O\left (\sum _{l_3 \leq Y} |{Q( {{{\bf{{m}}}}}, l_3)}| {\sum _{k^{ * } \leq Y / l_3}} A^{\omega (k^{ * })} k^{ * \frac {1}{2} n +1}\right ) \\ &= O\left (Y^{({1}/{2}) n +2} X^{\epsilon } \sum _{{l_3 \leq Y}} \frac { |{Q( {{{\bf{{m}}}}}, l_3)}| }{l_3^{({1}/{2}) n +2}}\right ). \end {array}\] Consequently, with $$n=8$$ and the notation in Lemma 21,   \[\begin {array}{rl} \Psi _3 (Y) &= O\left (\frac {X^{\epsilon }}{Y^{({1}/{2}) n -2}} \sum _{l_3 \leq Y} \frac {1}{l_3^{({1}/{2}) n +2}} {\sum _{\substack { {{{\bf{{m}}}}} \neq 0 \\ {\mathrm {grad}} F( {{{\bf{{m}}}}}) = 0}}} |{Q( {{{\bf{{m}}}}}, l_3)}| J( {{{\bf{{m}}}}}, Y)\right ) \\ &= O\left (\frac {X^{\epsilon }}{Y^2} \sum _{l_3 \leq Y} \frac { {{\mathrm {P}}} (Y, l_3)}{l_3^{({1}/{2}) n +2}}\right ) = O\left (\frac {X^{\epsilon } K(Y)}{Y^2}\right ) \quad {\mathrm {say}}. \end {array}\] (95) Noting that obviously $$ {{\mathrm {P}}} (Y, l_3)$$ does not exceed $$\Delta (Y, l_3)$$ in (63), we shall use (81) or (64) to estimate the summand in $$K(Y)$$ according as which is more favourable for values of $$l_3$$ satisfying   \[k_3 \leq Y^{1 - \delta } = Y_1 \quad {\mathrm {say}},\] (96) for a small positive number $$\delta$$, the criterion for which depends on a comparison between their second terms. But in the opposite case we shall use the simpler version of (82) given in the corollary of Lemma 21. Initiating the comparison between the above mentioned second terms by determining when they are essentially equal, we find that for $$\smash {l_3 = \prod _{{\alpha \geq 3}} k_{\alpha }}$$ we must have   \[Y^{({1}/{2}) n -1} N^{({1}/{2}) n +1} X^{-n} \prod _{\alpha \geq 3} \kappa _{\alpha }^{(({1}/{2}) n +1) \alpha } = X^{1- ({1}/{2}) n} {{\min}^{{({1}/{2}) n -1}}} (X,Y) \prod _{{\alpha \geq 3}} \kappa _{\alpha }^{\gamma _{\alpha , 2}}\] and hence that in this case   \[L = L(l_3) = \prod _{{\alpha \geq 3}} \kappa _{\alpha }^{\gamma _{\alpha , 2} - 5 \alpha } = \frac {Y^3 N^5}{X^5 {{\min}^{{3}}} (X,Y)} = N_1 \quad {\mathrm {say}},\] on assigning the value $$8$$ to $$n$$. The condition whether (81) or (64) be preferred when (96) holds being $$L \leq N_1$$ or $$L >N_1$$, we therefore can express $$K(Y)$$ through the inequality   \[\begin {array}{rl} K(Y) & \leq \sum _{\substack {L \leq N_1 \\ l_3 \leq Y}} \frac { {{\mathrm {P}}} (Y, l_3)}{l_3^{({1}/{2}) n +2}} + \sum _{\substack {L >N_1 \\ l_3 \leq Y; k_3 \leq Y_1}} \frac {\Delta (Y, l_3)}{l_3^{({1}/{2}) n +2}} + \sum _{\substack {l_3 \leq Y \\ k_3 >Y_1}} \frac { {{\mathrm {P}}} (Y, l_3)}{l_3^{({1}/{2}) n +2}} \\ &= K_1 (Y) +K_2 (Y) +K_3 (Y) \quad {\mathrm {say}}, \end {array}\] (97) the elements in which are to be estimated by separate methods. First, it now being suitable to let $$n=8$$ throughout, we have from Lemma 21 that   \[\begin {array}{rl} K_1 (Y) &= O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon }}{X^3} \sum _{{L \leq N_1}} \frac {G_2 (l_3)}{l_3^6}\right ) +O\left (\frac {Y^3 X^{\epsilon } N^{{5}/{2}}}{X^{{11}/{2}}} \sum _{{l_3 \leq Y}} \frac {G_1 (l_3)}{l_3^6}\right )\\ &= O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon }}{X^3} \sum _{1}\right ) +O\left (\frac {Y^3 X^{\epsilon } N^{{5}/{2}}}{X^{{11}/{2}}} \sum _{2}\right ) \quad {\mathrm {say}}. \end {array}\] (98) Next, the summand in $$\sum _{1}$$ is   \[\prod _{{\alpha \geq 3}} \kappa _{\alpha }^{\gamma _{\alpha ,2} - 6 \alpha }\] and therefore, for a suitable positive number $$\eta$$,   \[\sum _{1} \leq N_1^{\eta } \sum _{l_3 = 1}^{\infty } \prod _{{\alpha \geq 3}} \frac {1}{\kappa _{\alpha }^{\eta (\gamma _{\alpha ,2} - 5 \alpha ) - \gamma _{2, \alpha } +6 \alpha }} \,,\] (99) wherein the exponent of $$\kappa _{\alpha }$$ in the denominator is confirmed to be given in the following table in virtue of (83) and (84) If $$\eta = \frac {20}{27} + \epsilon$$, then not only all those exponents, including that in the least favourable case where $$\alpha = 8$$, exceed $$1$$ but also exceed a small positive multiple of $$\alpha$$ when $$\alpha$$ is large. Hence, by Euler's theorem on multiplicative functions, the series in (99) is convergent and we deduce that   \[\sum _{1} = O \left\{ \left (\frac {Y^3 N^5}{X^5 {{\min}^{{3}}} (X,Y)}\right )^{{20}/{27} + \epsilon }\right\} .\] (100) Also, somewhat similarly, the summand in $$\sum _{2}$$ is   \[\prod _{\alpha \geq 3} \frac {1}{\kappa _{\alpha }^{6 \alpha - \gamma _{1, \alpha }}},\] in which, by (83) and (84), the exponent $$6 \alpha - \gamma _{1, \alpha }$$ assumes the respective values $$\frac {3}{2}$$, $$\frac {3}{2}$$, $$2$$, $$\frac {1}{2}$$ for $$\alpha = 3, 4, 5, 6$$ and exceeds $$\frac {1}{4} \alpha$$ for $$\alpha >6$$. Therefore,   \[\sum _{2} \leq \sum _{{l_3 \leq Y}} \frac {1}{\kappa _3^{{3}/{2}} \kappa _4^{{3}/{2}} \kappa _5^{2} \kappa _6^{{1}/{2}} l_7^{{1}/{4}}} = O\left (\sum _{\kappa _6 \leq Y^{{1}/{6}}} \frac {1}{\kappa _6^{{1}/{2}}}\right ) = O\left (Y^{{1}/{12}}\right )\] (101) because   \[\sum _{l_7} \frac {1}{l_7^{{1}/{4}}}\] is convergent. $$\alpha$$  $$\eta (\gamma _{\alpha ,2} - 5 \alpha ) - \gamma _{2, \alpha } +6 \alpha$$  $$4$$  $$8 \eta -4$$  $$6$$  $$18 \eta -12$$  Odd  $$\dfrac {7}{2} \eta (\alpha -1) - \dfrac {5}{2} \alpha + \dfrac {7}{2}$$  Even and $$>6$$  $$\eta (\dfrac {7}{2} \alpha -1) - \dfrac {5}{2} \alpha +1$$  $$\alpha$$  $$\eta (\gamma _{\alpha ,2} - 5 \alpha ) - \gamma _{2, \alpha } +6 \alpha$$  $$4$$  $$8 \eta -4$$  $$6$$  $$18 \eta -12$$  Odd  $$\dfrac {7}{2} \eta (\alpha -1) - \dfrac {5}{2} \alpha + \dfrac {7}{2}$$  Even and $$>6$$  $$\eta (\dfrac {7}{2} \alpha -1) - \dfrac {5}{2} \alpha +1$$  View Large Secondly, by (97) and Lemma 16,   \[\begin {array}{rl} K_2 (Y) &= O\left (\frac {Y^3 X^{\epsilon } N^5}{X^8} \sum _{{L >N_1}} \frac {1}{l_3}\right ) +O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon }}{X^3} \sum _{{\substack {l_3 \leq Y \\ k_3 \leq Y_1}}} l_3^{{5}/{3}}\right ) \\ &= O\left (\frac {Y^3 X^{\epsilon } N^5}{X^8} \sum _{3}\right ) +O\left (\frac { {{\min}^{{3}}} (X,Y) X^{\epsilon }}{X^3} \sum _{4}\right ) \quad {\mathrm {say}}. \end {array}\] (102) Then, adopting the method of estimating $$\sum _{1}$$ to suit $$\sum _{3}$$, we choose a suitable positive number $$\eta _1$$ and find that   \[\sum _{3} \leq N_1^{- \eta _1} \sum _{l_3} \prod _{\alpha \geq 3} \frac {1}{\kappa _{\alpha }^{\alpha - \eta _1 (\gamma _{2, \alpha } - 5 \alpha )}},\] the exponent of $$\kappa _{\alpha }$$ therein being as follows: All these exponents exceed $$1$$ if $$\eta _1 = \frac {7}{27} - \epsilon$$ and, as in the derivation of (100), we therefore infer that   \[\sum _{3} = O\left (\left (\frac {Y^3 N^5}{X^5 {{\min}^{{3}}} (X,Y)}\right )^{-{7}/{27} + \epsilon }\right ).\] (103) Furthermore, turning to the other constituent in (102), we also have that   \[\begin {array}{rl} \sum _{4} & \leq Y^{{5}/{3}} \sum _{{\substack {k_3 l_4 \leq Y \\ k_3 \leq Y_1}}} 1 = Y^{{5}/{3}} \sum _{k_3 \leq Y_1} {\sum _{l_4 \leq Y / k_3}} 1 \\ &= O\left (Y^{{5}/{3} + {1}/{4}} \sum _{k_3 \leq Y_1} \frac {1}{k_3^{{1}/{4}}}\right ) = O\left (Y^{{5}/{3} + {1}/{4}} Y_1^{{1}/{12}}\right ) = O\left (Y^{2 - {\delta }/{12}}\right ). \end {array}\] (104) $$\alpha$$  $$\alpha - \eta _1 (\gamma _{2, \alpha } - 5 \alpha )$$  $$4$$  $$4 - 8 \eta _1$$  $$6$$  $$6 - 18 \eta _1$$  Odd  $$\alpha - \dfrac {7}{2} \eta _1 (\alpha -1)$$  Even and $$>6$$  $$\alpha - \eta _1 (\dfrac {7}{2} \alpha -1)$$  $$\alpha$$  $$\alpha - \eta _1 (\gamma _{2, \alpha } - 5 \alpha )$$  $$4$$  $$4 - 8 \eta _1$$  $$6$$  $$6 - 18 \eta _1$$  Odd  $$\alpha - \dfrac {7}{2} \eta _1 (\alpha -1)$$  Even and $$>6$$  $$\alpha - \eta _1 (\dfrac {7}{2} \alpha -1)$$  View Large Thirdly, it follows from Lemma 21 and its Corollary that   \[\begin {array}{rl} K_3 (Y) &= O\left (\frac {X^{\epsilon } Y^3 N^{{5}/{2}}}{X^{{11}/{2}}} \sum _{\substack {k_3 l_4 \leq Y \\ k_3 >Y_1}} \frac {1}{k_3^{{1}/{2}}}\right ) +O\left (\frac { X^{\epsilon } {{\min}^{{3}}} (X,Y)}{X^3} \sum _{{\substack {k_3 l_4 \leq Y \\ k_3 >Y_1}}} k_3^{{4}/{3}} l_4^{{5}/{2}}\right )\\ &= O\left (\frac {X^{\epsilon } Y^3 N^{{5}/{2}}}{X^{{11}/{2}}} \sum _{5}\right ) +O\left (\frac {X^{\epsilon } {{\min}^{{3}}} (X,Y)}{X^3} \sum _{6}\right ) \quad {\mathrm {say}}. \end {array}\] (105) Here,   \[\sum _{5} = \sum _{k_3 >Y_1} \frac {1}{k_3^{{1}/{2}}} {\sum _{l_4 \leq Y / k_3}} 1 = O\left (Y^{{1}/{4}} \sum _{k_3 >Y_1} \frac {1}{k_3^{{3}/{4}}}\right ) = O({Y^{{1}/{4}}) Y_1^{-{5}/{12}}} = O({Y^{- {1}/{6} + ({5}/{12}) \delta }})\] (106) and   \[\sum _{6} \leq Y^{{4}/{3}} \sum _{{\substack {k_3 l_4 \leq Y \\ k_3 >Y_1}}} l_4^{{7}/{6}} = O\left (Y^{{4}/{3} + {17}/{12}} \sum _{k_3 >Y_1} \frac {1}{k_3^{{17}/{12}}}\right ) = O({Y^{{11}/{4}} Y_1^{- {13}/{12}}}) = O({Y^{{5}/{3} + ({13}/{12}) \delta }}),\] (107) with which estimates we complete the treatment of all the ingredients that $$\Psi _3 (Y)$$ contains. The terms in (98) and (102) containing $$\sum _{1}$$ and $$\sum _{3}$$, respectively, are seen through (100) and (103) to be   \[O \left\{\frac {X^{\epsilon } {{\min}^{{3}}} (X,Y)}{X^3} \left (\frac {N^5 Y^3}{X^5 {{\min}^{{3}}} (X,Y)}\right )^{{20}/{27}}\right\} \quad {\mathrm {and}}\quad O \left\{\frac {X^{\epsilon } Y^3 N^5}{X^8} \left (\frac {N^5 Y^3}{X^5 {{\min}^{{3}}} (X,Y)}\right )^{- {7}{27}}\right\},\] which, being formally equal, coalesce by (97) into a contribution of   \[O \left\{X^{\epsilon } \left (\frac {N^5 Y^3}{X^8}\right )^{{20}/{27}}\right\} = O\left (\frac {X^{\epsilon } Y^{{20}/{9}}}{X^{{10}/{27}}}\right )\] (108) to $$K(Y)$$ regardless of whether $$Y <X$$ or $$Y \geq X$$. Secondly, from the estimates (101) and (106), there is the contribution   \[O\left (\frac {X^{\epsilon } Y^3}{X^{{7}/{4}}} ({Y^{{1}/{12}} +Y^{- {1}/{6} + ({5}/{12}) \delta })}\right )\] (109) to $$K(Y)$$, while that from the estimates (103) and (107) for $$\sum _{4}$$ and $$\sum _{6}$$ is   \[ {O \left({{\frac {X^{\epsilon } {{\min}^{{3}}} (X,Y)}{X^3} (Y^{2 - ({1}/{12}) \delta } +Y^{{5}/{3} + ({13}/{12}) \delta })}}\right)}\] Hence, adding this to (108) and (109) with a suitably small value of $$\delta$$ to bound $$\Psi _3 (Y)$$ in (95), we conclude that   \[\Psi _3 (Y) = {O \left({{\frac {X^{\epsilon } Y^{{2}/{9}}}{X^{{10}/{27}}}}}\right)} + {O \left({{\frac {X^{\epsilon } Y^{{13}/{12}}}{X^{{7}/{4}}}}}\right)} + {O \left({{\frac {X^{\epsilon } {{\min}^{{3}}} (X,Y)}{X^3 Y^{{\delta }/{12}}}}}\right)}\] for $$Y \leq A N$$. Consequently, in the usual way, we derive the estimate   \[\begin {array}{rl} \Upsilon _3 (X) &= {O \left({{\frac {X^{\epsilon } N^{{2}/{9}}}{X^{{10}/{27}}}}}\right)} + {O \left({{\frac {X^{\epsilon } N^{{13}/{12}}}{X^{{7}/{4}}}}}\right)} + {O \left({{\frac {X^{\epsilon }}{X^{{\delta }/{12}}}}}\right)} + {O \left({{\frac {X^{\epsilon }}{X^{{\delta }/{12}}}}}\right)} \\ &= {O \left({{\frac {X^{\epsilon }}{X^{{1}/{27}}}}}\right)} + {O \left({{\frac {X^{\epsilon }}{X^{{1}/{8}}}}}\right)} + {O \left({{\frac {X^{\epsilon }}{X^{{\delta }/{12}}}}}\right)} \\ &= {o (1)} \quad (X \to \infty ) \end {array}\] (110) that we sought. 16. Calculation of $$\Upsilon _1 (X)$$ Associated with the sum $$\Upsilon _1 (X)$$ appearing in (14) there is the formal singular series   \[{{\frak {S}}} = \sum _{k=1}^{\infty } \frac {Q (0,k)}{k^n},\] which for $$n \geq 7$$ was fully investigated in §19 of our Ib. Extracting what we need from this paper when $$n=8$$, we have the bound   \[\frac {Q (0,k)}{k^8} = {O \left({{\frac {A^{\omega (k)}}{k^{{5}/{3}}}}}\right)}\] (111) with the consequential absolute convergence of the series, whence   \[{{\frak {S}}} = \zeta (5) \prod _{p} \frac {\nu ^{\prime } (p^{2 \gamma +1})}{p^{7(2 \gamma +1)}} \quad (p^{\gamma } {{\parallel } } D)\] in the notation of § 6. Thus, $${{\frak {S}}} >0$$ since $$f( {{{\bf{{x}}}}})$$ is assumed to have a non-trivial zero in $${\Bbb {Q}}_p$$ and therefore one in $${\Bbb {Z}}_p$$ that is indivisible by $$p$$. Next, having reminded ourselves that $$ {{{\bf{{a}}}}}$$ is a sufficiently large scalar multiple of a given real (non-singular) zero of $$f( {{{\bf{{x}}}}})$$, we must turn to the integral   \[I_{k} (0) = \int _{ {{{\bf{{a}}}}} - {\boldsymbol {{\upsilon }}} }^{ {{{\bf{{a}}}}} + {\boldsymbol {{\upsilon }}} } \Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}}) h\left (\frac {k}{N}, f( {{{\bf{{t}}}}})\right ) {d} {{{\bf{{t}}}}}\] and subject it to a simplification of Heath-Brown's treatment that suffices for our current needs. Since $${\mathrm {grad}} f( {{{\bf{{a}}}}}) \neq 0$$, we may assume by way of a typical example that $$|{ {\partial {^{{}}}{f} / \partial {{t_1}^{{}}}} }| >A$$ within the region of integration and may therefore replace the variables of integration by $$y = f( {{{\bf{{t}}}}})$$ and those in $$ {{{\bf{{t}}}}} ^{\prime } = (t_2, \ldots , t_n)$$. Thus, taking $$y$$ to be the outermost variable of integration, we can express the integral as   \[\int h(r,y) {\mathscr {S}} (y) \, {d} y,\] (112) where the region of integration in   \[{\mathscr {S}} (y) = \int \frac {\Gamma ( {{{\bf{{t}}}}} - {{{\bf{{a}}}}})}{ {\partial {^{{}}}{f} / \partial {{t_1}^{{}}}} } {d} {{{\bf{{t}}}}} ^{\prime }\] is over all $$ {{{\bf{{t}}}}} ^{\prime }$$ compatible with the conditions $$||{ {{{\bf{{t}}}}} ^{\prime } - {{{\bf{{a}}}}} ^{\prime }}|| \leq 1$$ and $$y = f( {{{\bf{{t}}}}})$$. We now apply Heath-Brown's important Lemma 9 of Hb to (112). This gives   \[I_k (0) = {\mathscr {S}} (0) + {O \left({{\frac {k^{{1}/{3}}}{N^{{1}/{3}}}}}\right)} = O(1)\] with the result through (111) that   \[\begin {array}{rl} \Upsilon _1 (X) &= {\mathscr {S}} (0) {{\frak {S}}} + {O \left({{\sum _{k >A_1 N} \frac {A^{\omega (k)}}{k^{{5}/{3}}}}}\right)} + {O \left({{\frac {1}{N^{{1}/{3}}} \sum _{k \leq A_1 N} \frac {A^{\omega (k)}}{k^{{4}/{3}}}}}\right)} \\ &= {\mathscr {S}} (0) {{\frak {S}}} + {O \left({{\frac {\log ^{A-1} N}{N^{{2}/{3}}}}}\right)} + {O \left({{\frac {1}{N^{{1}/{3}}}}}\right)} \\ &= {\mathscr {S}} {{\frak {S}}} + {o (1)} \quad {\mathrm {say}}, \end {array}\] (113) as $$X \to \infty$$. Also, since $$f( {{{\bf{{a}}}}}) = 0$$, we know by the implicit function theorem that there is a positive number $$\delta <\frac {1}{2}$$ with the property that there is a zero $$t_1$$ of $$f (t_1, {{{\bf{{t}}}}} ^{\prime })$$ satisfying $$|{t_1 - a_1}| <\delta$$ whenever $$||{ {{{\bf{{t}}}}} ^{\prime } - {{{\bf{{a}}}}} ^{\prime }}|| <\delta$$. It therefore follows that   \[{\mathscr {S}} >0.\] (114) 17. The theorem At last the theorem is in hand. From (14), (94), (110), and (113) we infer that   \[\Upsilon (X) \geq \frac {1}{2} X^5 ({\mathscr {S}} {{\frak {S}}} + {o (1)})\] and then from (114) the inequality   \[\Upsilon (X) >A X^5,\] which implies that the indeterminate equation $$f( {{{\bf{{l}}}}}) = 0$$ has a non-zero solution. This has been obtained on the assumption (5), the negation of which has already been shown to imply the existence of such a solution. We therefore have proved our theorem: Theorem Let $$f( {{{\bf{{x}}}}})$$ be a non-singular octonary cubic form with rational integral coefficients. Then on Hypothesis HW the indeterminate equation   \[f( {{{\bf{{l}}}}}) = 0\] has a non-trivial integral solution if $$f( {{{\bf{{x}}}}})$$ have a non-trivial zero in every $$p$$-adic field $${\Bbb {Q}}_p$$. 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TI - On octonary cubic forms
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/pdt066
DA - 2014-02-24
UR - https://www.deepdyve.com/lp/oxford-university-press/on-octonary-cubic-forms-aBSFTC2sVx
SP - 241
EP - 281
VL - 109
IS - 1
DP - DeepDyve
ER -