# Sign Changes of Kloosterman Sums with Almost Prime Moduli. II

Sign Changes of Kloosterman Sums with Almost Prime Moduli. II Abstract We prove that the Kloosterman sum $$S(1,1;c)$$ changes sign infinitely often as $$c$$ runs over squarefree numbers with at most 7 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler, Matomäki, and the author. 1 Introduction We continue our study on sign changes of Kloosterman sums, which can be defined by   S(m,n;c)=∑∑∗amodc⁡e(ma+na¯c) for each positive integer $$c$$ and integers $$m,n$$, where $$a\overline{a}\equiv1\bmod c.$$ Let us state our theorem quickly. Theorem 1.1. There exists an absolute constant $$c_0>0$$ such that for sufficiently large $$X>0$$,   |{c⩽X:S(1,1;c)≷0, ω(c)⩽7, c squarefree}|⩾c0Xlog⁡X, where $$\omega(c)$$ denotes the number of distinct prime factors of $$c$$. □ In view of the Weil bound   |S(m,n;p)|⩽2p,  (m,n,p)=1 (1.1) for each prime $$p$$, the main motivation of Theorem 1.1 is the Sato–Tate conjecture of Katz [9], which asserts that the normalized Kloosterman sums $$S(1,1,p)/2\sqrt{p}$$ become equidistributed with respect to the Sato–Tate measure as $$p$$ runs over all the primes; it would then follow immediately that $$S(1,1;p)$$ changes sign infinitely often as $$p$$ varies. The study on sign changes was initiated by Fouvry and Michel [5, 6], who obtained a weaker constant 23 by a pioneering combination and application of the Selberg sieve, spectral theory of automorphic forms and $$\ell$$-adic cohomology. The subsequent improvements are due to Sivak-Fischler [14], Matomäki [12], and the author [15], where the relevant constants are 18, 15, and 10, respectively. As in previous works, the proof of Theorem 1.1 also utilizes the Selberg sieve, and the framework is based on that in [15]. The main ingredient in this paper is the introduction of a truncated divisor function and a successful application of Sato–Tate distributions of Kloosterman sums in the vertical direction. The framework of proof will be outlined in Section 2. 1.1 Notation and convention As usual, $$\mu,\varphi$$ denote the Möbius and Euler functions, respectively. The superscript $$*$$ in summation indicates primitive elements. We use $$\varepsilon$$ to denote a sufficiently small positive number and $$A$$ a reasonably large number, which are not to be the same on each occasion. For $$\alpha,\beta>0$$, define a truncated divisor function   τ(n;α,β)=∑dl=nd<lp|d⇒p>(logn)Aαω(d)βω(l). We also define   σ−1∗(n)=∑d|n,d>11d. Throughout this paper, we assume $$g(x)$$ is a fixed non-negative smooth function supported in $$[1,2]$$. and its Mellin transform is defined as   g~(s)=∫0+∞g(x)xs−1dx. Integrating by parts, we have   g~(s)≪(|s|+1)−A for any $$A\geqslant0.$$ Moreover, we input the normalization   g~(1)=∫0+∞g(x)dx=1. 2 Outline of the proof We prove Theorem 1.1 by applying the Selberg sieve. Let $$\lambda=(\lambda_d)$$ be the Selberg sieve weight of level $$\sqrt{D}:=X^{1/4}\exp(-\sqrt{\log X})$$. We would like to start from the following weighted sum   H±(X)=∑ng(nX)|S(1,1;n)|±S(1,1;n)nμ2(n){ρ−τ(n;α,β)}(∑d|nλd)2, (2.1) where $$\rho,\alpha,\beta>0$$ are three parameters to be chosen later.1 Upon suitable choices of $$\rho,\alpha,\beta$$, and $$(\lambda_d)$$, the positivity of $$H^\pm(X)$$ implies, for $$X$$ large enough, that there exists $$n\in(X,2X]$$ with   τ(n;α,β)<ρ for which $$S(1,1;n)\gtrless0$$. This will establish Theorem 1.1 from a careful analysis on the combinatorics of $$\tau(n;\alpha,\beta)$$. From (2.1), we have   H±(X)⩾ρH1(X)−2H2(X)±ρH3(X), where   H1(X)=∑ng(nX)|S(1,1;n)|nμ2(n)(∑d|nλd)2,H2(X)=∑ng(nX)|S(1,1;n)|nμ2(n)τ(n;α,β)(∑d|nλd)2,H3(X)=∑ng(nX)S(1,1;n)nμ2(n)(∑d|nλd)2. The lower bound for $$H^\pm(X)$$ follows from a lower bound for $$H_1(X)$$, an upper bound for $$H_2(X)$$, and a good control on the order of magnitude of $$H_3(X)$$. Let $$F(x)$$ be a fixed smooth function supported on $$[0,1]$$ and vanishes at 0 to a suitable order. We choose $$(\lambda_d)$$ such that   λd=μ(d)F(log⁡(D/d)log⁡D),  d⩽D. (2.2) The following lower bound for $$H_1(X)$$ was obtained in [12] and [15] for special choices of $$F$$. The general case can be treated similarly and we state the following proposition without proof. Proposition 2.1. For any sufficiently large $$X$$, we have   H1(X)⩾g~(1)∑2⩽i⩽52iAi(F)Ci⋅Xlog⁡X(1+o(1)), where $$C_2= 0.11109,\ C_3= 0.03557,\ C_4= 0.01184,\ C_5= 0.00396,$$  Ai(F)=∫⋯∫RiLi2(γ,F;X1−α2−⋯−αi,Xα2,…,Xαi)α2⋯αi(1−α2−⋯−αi)dα2⋯dαi,Li(F;α1,α2,…,αi)=∑A⊆{α1,α2,…,αi}∑α∈Aα<14(−1)|A|F(1−4∑α∈Aα), and   R2:={α2∈[η,1):(34+η)(1−α2)<α2<12},R3:={(α2,α3)∈[η,1)2:12(1−α2−α3)<α2,  α3<α2<1−α2−α3},R4:={(α2,α3,α4)∈[η,1)3:12(1−α2−α3−α4)<α2+α3}     ∩{(α2,α3,α4)∈[η,1)3:α4<α3<α2<1−α2−α3−α4},R5:={(α2,α3,α4,α5)∈[η,1)4:12(1−α2−α3−α4−α5)<α2+α3+α4}     ∩{(α2,α3,α4,α5)∈[η,1)4:12(α3+α4+α5)<α2}     ∩{(α2,α3,α4,α5)∈[η,1)4:α5<α4<α3<α2<1−α2−α3−α4−α5},η:=10−2016. Proposition 2.2. Let $$k$$ be a positive integer with $$\alpha=3\pi k/16,\beta=k/4$$, and $$\mathfrak{c}(k,F)$$ a constant defined by (4.6). For any sufficiently large $$X,$$ we have   H2(X)⩽2g~(1)⋅c(k,F)Xlog⁡X(1+o(1)). □ The proof of Proposition 2.2 will be presented in Section 4. The constant $$\frac{8}{3\pi}$$ comes from certain Sato–Tate distributions of Kloosterman sums in the vertical direction. The following proposition follows from the Bombieri–Vinogradov type estimate for Kloosterman sums that was initiated by Fouvry and Michel [6] deriving from the spectral theory of automorphic forms; the current version can be found in [15, Lemma 9]. Proposition 2.3. For any $$A>0$$ and sufficiently large $$X$$, we have   H3(X)≪X(log⁡X)−A, where the implied constant depends on $$A$$ and $$g$$. □ Theorem 1.1 follows from suitable choices of all parameters appearing in Propositions 2.1 and 2.2. The details, including the partial optimization on the choice of $$F$$, will be given in Section 6. 3 Auxiliary lemmas 3.1 Distribution in arithmetic progressions The following lemma is an extension of the classical Barban–Davenport–Halberstam theorem, which is originally devoted to primes in arithmetic progressions. The statement and proof can be found in [1, Theorem 0]. Lemma 3.1. Let $$(\alpha_n),n\leqslant N,$$ be any sequences satisfying the “Siegel–Walfisz” condition:   ∑n≡amodq(n,d)=1αn−1φ(q)∑(n,qd)=1αn≪τ(d)O(1)‖α‖N1/2(log⁡N)−C for any $$q\geqslant1,d\geqslant1,C>0,(q,a)=1,a\neq0$$ with an implied constant in $$\ll$$ depending only on $$C$$. Here $$\|\cdot\|$$ denotes the $$\ell_2$$-norm. Then, for any $$A>0,$$ there exists certain $$B=B(A)>0$$ such that   ∑q⩽Q∑∑∗amodq⁡|∑n≡amodqαn−1φ(q)∑(n,q)=1αn|2≪‖α‖2N(log⁡N)−A, provided that $$Q\leqslant N(\log N)^{-B},$$ where the implied constant depends only on $$A.$$ □ 3.2 Partition of unity We require the following lemma as a smooth partition of unity, which enables us to separate variables attaching smooth weights (see [4, Lemme 2], for instance). Lemma 3.2. For every $$\varDelta>1,$$ there exists a sequence $$\{b_{l,\varDelta}\}_{l\geqslant0}$$ of smooth functions with support included in $$[\varDelta^{l-1},\varDelta^{l+1}]$$, such that   ∑l⩾0bl,Δ(x)=1  for all x⩾1, and   xjbl,Δ(j)(x)≪(1−Δ−1)−j  for all x⩾1 and l⩾0. (3.1) □ 3.3 Mellin inversions We state two results from complex analysis as typical applications of the Mellin inversion formula. Lemma 3.3. Suppose $$k$$ is a non-negative integer. Then   k!2πi∫1−i∞1+i∞xssk+1ds={0,   0<x⩽1,(log⁡x)k,x>1, holds for any fixed positive number $$x.$$ □ Let $$P$$ be a smooth function with $$P(0)=0$$, and suppose it admits the following Taylor expansion   P(x)=∑k⩾0akk!xk. For $$N>1$$ and $$s\in\mathbf{C}\setminus\{0\}$$, define   PˇN(s)=∑k⩾0ak(slog⁡N)k. (3.2) Lemma 3.4. Suppose $$N>1$$ is not an integer. For any coefficients $$y_n$$ with $$y_n=O(\tau(n)^{O(1)}(\log n)^{O(1)}),$$ we have   ∑n⩽NynP(log⁡(N/n)log⁡N)=12πi∫2−i∞2+i∞PˇN(s)Y(s)Nssds, where   Y(s)=∑n⩾0ynns,   ℜs>1. □ Proof This is a consequence of [11, Lemma 2.1]. ■ 3.4 Sato–Tate distribution of Kloosterman sums From the works of Deligne [3] and Katz [10], it follows that the map   m↦S(m,1;p)p=2cos⁡θp(m),  m∈Fp× is the trace function of an $$\ell$$-adic sheaf $$\mathcal{K}l$$ on $$\mathbf{G}_{m}(\mathbf{F}_p)=\mathbf{F}_p^\times$$, which is of rank 2 and pure of weight 0. Therefore,   2cos⁡θp(m)=tr(Frobm,Kl). A celebrated theorem of Katz states that the set $$\{ \theta_p(m):m \in\mathbf{F}_p^\times\}$$ becomes equidistributed with respect to the Sato–Tate measure $$\mu_{\rm ST}=\frac{2}{\pi}\sin\theta\mathrm{d}\theta$$ as $$p \rightarrow \infty$$. By the Weyl equidistribution criterion and the Peter–Weyl theorem, the proof of Katz’s Sato–Tate distribution reduces to the study of   ∑m∈Fp×symk(θp(m))=∑m∈Fp×tr(Frobm,symkKl), where $$\mathrm{sym}^k\mathcal{K}l$$ is the $$k$$th symmetric power of the Kloosterman sheaf $$\mathcal{K}l$$ (i.e., the composition of the sheaf $$\mathcal{K}l$$ with the $$k$$th symmetric power representation of $$SL_2$$) and   symk(θ)=sin⁡(k+1)θsin⁡θ. Then the equidistribution of Katz can be formulated equivalently as   |∑m∈Fp×symk(θp(m))|⩽12(k+1)p; (3.3) we refer to Example 13.6 and the preceding theorem in [10] for more details. In fact, the square-root cancellation also holds if replacing $$\theta_p(m)$$ by $$\theta_p(f(m))$$ for any non-constant rational function $$f$$ of fixed degree over $$\mathbf{F}_p^\times,$$ in which case the independence of the upper bound on $$k$$ is polynomial of higher degree. In particular, it was proved by Michel [13], for each fixed $$k\geqslant1$$, that   ∑m∈Fp×symk(θp(m¯2))≪kO(1)p, (3.4) where the implied constant is absolute. The sheaf here should be $$\mathrm{sym}^k([-2]^*\mathcal{K}l)$$. In fact, such square-root cancellation follows from the Riemann Hypothesis of Deligne [3] on the quasi-orthogonality of trace functions of geometrically irreducible sheaves. One can also refer to a formulation of Fouvry, Kowalski and Michel that is more accessible to analytic applications, see the survey [8, Theorem 4.1], for instance. In particular, we may state the following square-root cancellations as consequences of the Riemann Hypothesis of Deligne. Lemma 3.5. Let $$\chi,\psi$$ be multiplicative and additive characters modulo $$p$$, respectively. For each fixed positive integer $$k,$$ we have   ∑∑∗mmodp⁡χ(m)symk(θp(m¯2))≪kO(1)p, (3.5) and   ∑∑∗mmodp⁡ψ(m)symk(θp(m¯2))≪kO(1)p. (3.6) Suppose $$(n_1n_2,p)=1$$ and $$n_1\not\equiv\pm n_2\bmod p.$$ For each fixed positive integers $$k_1,k_2,$$ we have   ∑∑∗mmodp⁡ψ(m)symk1(θp(n1m¯2))symk2(θp(n2m¯2))≪(k1k2)O(1)p. (3.7) The above implied constants are all absolute. □ For $$(n,c)=1$$, put   K(n,c)=S(n¯2,1;c)c. (3.8) It follows from the Chinese Remainder Theorem that $$K(n,rs)=K(nr,s)K(ns,r)$$ for all $$n,r,s$$ with $$(r,s)=(n,rs)=1.$$ We also have the following correlations of Kloosterman sums. Lemma 3.6. For all $$n_1,n_2,h$$ with $$(n_1n_2,p)=1,$$ we have   ∑∑∗amodp⁡|K(n1a,p)||K(n2a,p)|e(hap)≪p1/2(h,p)1/2, where the implied constants are absolute. □ Proof If $$n_1\equiv \pm n_2\bmod p$$, the lemma follows from (3.6) and the expansion   |x|2=14+14sym2(x). The case of $$n_1\not\equiv\pm n_2\bmod p$$ is a consequence of (3.7) and the expansion (3.9)   |x|=43π+∑k⩾1βksym2k(x) (3.9) with $$\beta_k=O(1/k!)$$ from the theory of Chebyshev polynomials. ■ The following first moment of Kloosterman sums follows from (3.4) and the expansion (3.9). Lemma 3.7. For sufficiently large prime $$p,$$ we have   ∑∑∗amodp⁡|K(a,p)|=83πp+O(p). □ We also require the following evaluation of bilinear sums of Kloosterman sums. Lemma 3.8. Let $$q$$ be a squarefree number and $$K$$ defined by (3.8). For any bounded real coefficients $$(a_m)$$ and $$(b_n)$$ with supports in $$(M,2M],(N,2N],$$ respectively, we have   ∑∑(mn,q)=1⁡ambn|K(mn,q)|=1φ(q)∑∑∗rmodq⁡|K(r,q)|∑∑(mn,q)=1⁡ambn+O(‖A‖‖B‖(MN)1/2(N−1/2qε+M−1/2q1/4+ε+3ω(q)σ−1∗(q)1/2)), where $$\|\cdot\|$$ denotes the $$\ell_2$$-norm and the implied constant depends only on $$\varepsilon.$$ □ Remark 1. The appearance of $$3^{\omega(q)}\sigma_{-1}^*(q)^{1/2}$$ in the error term shows that one cannot obtain the desired asymptotic evaluation if $$q$$ has small prime factors, in which case the advantage of bilinear forms becomes inapparent. While $$q$$ is a large prime, the Lemma 3.8 is an immediate consequence of the bilinear form estimates involving $$\mathrm{sym}_k(\theta_p(\overline{mn}^2))$$ following the approach of Pólya–Vinogradov method, and a relevant study (with slightly stronger estimates) can be found for instance in [13] and [7]. Nevertheless, Lemma 3.8 gives a typical asymptotic formula as long as $$N>p^{2\varepsilon}$$, $$M>p^{1/2+2\varepsilon}$$, and $$9^{\omega(q)}\sigma_{-1}^*(q)=o(1)$$. In particular, the last restriction holds on average if we assume $$q$$ has no prime factors below $$(\log q)^{2016}.$$ □ Proof We prove this lemma by the dispersion method. Put   Δ(A,B;q)=∑∑(mn,q)=1⁡ambn|K(mn,q)|−|A||B|φ(q)∑∑∗rmodq⁡|K(r,q)|, where   |A|=∑(m,q)=1am,   |B|=∑(n,q)=1bn. Thus   Δ(A,B;q)=∑∑∗rmodq⁡|K(r,q)|(∑∑mn≡rmodq⁡ambn−1φ(q)∑∑(mn,q)=1⁡ambn)=∑(m,q)=1αm∑∑∗rmodq⁡|K(r,q)|(∑n≡m¯rmodqbn−1φ(q)∑(n,q)=1bn). By Cauchy inequality, we have   Δ(A,B;q)2⩽‖A‖2Δ(B), (3.10) where   Δ(B)=∑(m,q)=1f(m){∑∑∗rmodq⁡|K(r,q)|(∑n≡m¯rmodqbn−1φ(q)∑(n,q)=1bn)}2. Here $$f$$ is a non-negative smooth function with compact support, containing $$(M,2M]$$, where $$f$$ takes value 1. Squaring out and switching summation, we may write   Δ(B)=Δ1(B)−2Δ2(B)+Δ3(B) with   Δ1(B)=∑∑(n1n2,q)=1⁡bn1bn2∑∑∗⁡∑∑∗r1,r2modqr1n2≡r2n1modq⁡|K(r1,q)||K(r2,q)|∑m≡n1¯r1modqf(m),  Δ2(B)=|B|φ(q)∑(n,q)=1bn∑∑∗⁡∑∑∗r1,r2modq⁡|K(r1,q)||K(r2,q)|∑m≡n¯r1modqf(m), and   Δ3(B)=|B|2φ(q)2(∑∑∗rmodq⁡|K(r,q)|)2∑(m,q)=1f(m). By Poisson summation,   Δ1(B)=1q∑h∈Zf^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗⁡∑∑∗r1,r2modqr1n2≡r2n1modq⁡|K(r1,q)||K(r2,q)|e(r1hn1¯q)=1q∑h∈Zf^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|e(rhq)=Δ11(B)+Δ12(B), where   Δ11(B)=f^(0)q∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|, and   Δ12(B)=1q∑h≠0f^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|e(rhq). From the orthogonality of Dirichlet characters, we derive that   Δ11(B)=f^(0)qφ(q)∑χmodq|∑nbnχ(n)|2|∑∑∗rmodq⁡χ(r)|K(r,q)||2=f^(0)qφ(q)∑dl=q∑∑∗χmodd⁡|∑(n,q)=1bnχ(n)|2|∑rmodq(r,q)=1χ(r)|K(r,q)||2, where $$\sideset{}{^*}\sum_\chi$$ denotes the summation over primitive characters $$\chi\bmod d$$. For $$dl=q$$ and $$(r,q)=1$$, we have $$K(r,q)=K(rd,l)K(rl,d)$$, which yields   ∑rmodq(r,q)=1χ(r)|K(r,q)|=χ¯(l)∑∑∗r1modd⁡χ(r1)|K(r1,d)|⋅∑∑∗r2modl⁡|K(r2,l)| for each primitive character $$\chi\bmod d.$$ The term with $$d=1$$ gives the contribution   f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2, which is expected to dominate the asymptotic evaluation of $$\Delta_{11}(\mathcal{B}).$$ The remaining contributions from $$d>1$$ are at most   ⩽f^(0)qφ(q)∑dl=qd>1l24ω(l)∑∑∗χmodd⁡|∑(n,q)=1bnχ(n)|2|∑∑∗rmodd⁡χ(r)|K(r,d)||2. The inner most sum over $$r$$ is at most $$O(d^{1/2}3^{\omega(d)})$$ in view of (3.5) and the Chinese Remainder Theorem. Thus the above quantity is bounded by   ≪f^(0)qφ(q)9ω(q)∑dl=qd>1dl2∑χmodd|∑(n,q)=1bnχ(n)|2≪‖B‖2M9ω(q)qφ(q)∑dl=qd>1dl2(d+N)≪‖B‖2Mqε+‖B‖2M N9ω(q)σ−1∗(q). We thus obtain that   Δ11(B)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O((qεM+MN9ω(q)σ−1∗(q))‖B‖2). From integration by parts, it follows that   f^(λ)≪M(1+M|λ|)−A for any fixed $$A\geqslant0$$. Thus,   Δ12(B)=1q∑|h|<q1+ε/Mf^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|e(rhq)+O(1). In view of Lemma 3.6 and the Chinese Remainder Theorem, we derive that   Δ12(B)≪Mq∑|h|<q1+ε/M∑∑(n1n2,q)=1⁡|bn1|2q1/2+ε(h,q)1/2≪q1/2+εN‖B‖2. Therefore, we obtain   Δ1(B)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O(qε(M+q1/2N)‖B‖2). Similarly, we have   Δ2(B)=|B|qφ(q)∑h∈Zf^(hq)∑(n,q)=1bn∑∑∗⁡∑∑∗r1,r2modq⁡|K(r1,q)||K(r2,q)|e(r1hn¯q)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O(q1/2+εN‖B‖2), and   Δ3(B)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O(qεN‖B‖2). Collecting the above evaluations, we obtain   Δ(B)≪(q1/2+εN+qεM+MN9ω(q)σ−1∗(q))‖B‖2, from which and (3.10), Lemma 3.8 follows. ■ 4 Proof of Proposition 2.2: Upper bound for $$\boldsymbol{H}_{\boldsymbol{2}}\boldsymbol{(X)}$$ Recall the definition (3.8) of $$K(m,n)$$ and the twisted multiplicativity, then $$H_2(X)$$ becomes   H2(X)=∑ng(nX)|K(1,n)|μ2(n)τ(n;α,β)(∑d|nλd)2=∑∑m<np|m⇒p>(log⁡m)A⁡g(mnX)|K(m,n)||K(n,m)|μ2(mn)αω(m)βω(n)(∑d|mnλd)2. From the Weil bound for Kloosterman sums, it follows that   H2(X)⩽∑∑m<np|m⇒p>(log⁡m)A⁡g(mnX)|K(n,m)|μ2(mn)αω(m)(2β)ω(n)(∑d|mnλd)2. Following Lemma 3.2, we may separate variables $$m,n$$ by introducing a series of smooth functions of the shape $$b_{l,\varDelta}$$. To do so, we put   Δ=1+(log⁡X)−B for some parameter $$B\geqslant1.$$ Denote by $$M,N$$ some parameters of the form   M=Δl1=(1+(log⁡X)−B)l1,   N=Δl2=(1+(log⁡X)−B)l2 (4.1) for some $$l_1,l_2\in\mathbf{N}.$$ For the variables $$M=\varDelta^{l_1},N=\varDelta^{l_2}$$, we define   U=Ul1=bl1,Δ,   V=Vl2=bl2,Δ (4.2) as given in Lemma 3.2. Thus, the derivatives of $$U$$ and $$V$$ satisfy   xlU(l)(x),  xlV(l)(x)≪(log⁡X)Bl (4.3) for all $$l\geqslant 0$$, where the implied constant may depend on $$l$$. Moreover, the integration by parts gives   U^(ξ)≪M(log⁡X)Bl(1+|ξ|M)−l,   V^(ξ)≪N(log⁡X)Bl(1+|ξ|N)−l (4.4) for any fixed $$l\geqslant0$$, and the implied constant also depends on $$l$$. For $$\mathbf{H}=(M,N)$$, we consider the following sum   H2(X,H)=∑∑m<np|m⇒p>(log⁡m)A⁡U(m)V(n)g(mnX)|K(n,m)|μ2(mn)αω(m)(2β)ω(n)(∑d|mnλd)2, where $$U,V$$ are two smooth functions defined as above, so that   H2(X)⩽∑HH2(X,H). The above summation is restricted to at most $$O((\log X)^{2B})$$ tuples of $$\mathbf{H}=(M,N)$$ with $$M,N$$ given by (4.1). From the support of $$g$$ and the restriction that $$m<n$$, we may assume   MN≍X,   M≪N. Let $$\delta>0$$ be a sufficiently small number. For $$N>X^{1/2+2\delta}$$, we write   H2(X,H)=∑l∑m≡0modlp|m⇒p>(log⁡m)AU(m)μ2(m)αω(m)Ψ(N;m,l), where   ξ(n)=∑[d1,d2]=nλd1λd2, (4.5) and   Ψ(N;m,l):=∑∑(nd,m)=1nd>m⁡V(nd)g(mndX)|K(nd,m)|μ2(nd)(2β)ω(nd)ξ(dl). We evaluate $$H_2(X;M,N)$$ in two complementary methods. Note that $$dl\leqslant D\leqslant X^{1/2}$$ in view of the level of Selberg sieve weights. We divide the sum over $$d$$ to $$d\leqslant X^\delta$$ and $$d>X^\delta$$, so that $$\Psi(N;m,l)$$ can be split to   Ψ(N;m,l)=∑n∑d⩽Xδ+∑n∑d>Xδ=Ψ1(N;m,l)+Ψ2(N;m,l), say. From the support of $$V$$ we see $$nd\asymp N>X^{1/2+2\delta}.$$ Thus, $$n\gg X^{1/4+2\delta}$$ if $$d^\delta<d<X^{1/4+\delta}$$, and $$n\gg X^{2\delta}$$ if $$X^{1/4+\delta}\leqslant d<X^{1/2}$$. Furthermore, we have $$X^{1/4+2\delta}>M^{1/2+\delta}$$. It then follows from Lemma (3.8) that   Ψ2(N;m,l)=1φ(m)∑∑∗amodm⁡|K(a,m)|∑∑(nd,m)=1d>Xδ,nd>m⁡V(nd)g(mndX)μ2(nd)(2β)ω(nd)ξ(dl)+O(N1−δ3+(log⁡M)−A/4). If $$d\leqslant X^\delta$$, we see $$n\gg X^{1/2+\delta}$$ from the support of $$V$$. Note that $$m\asymp M<(N/d)^{1-\delta}.$$ In such case we may arrange the contributions, coming from $$\Psi_1(N;m,l)$$, in $$H_2(X;M,N)$$ as   =∑d<Xδμ2(d)(2β)ω(d)∑p|m⇒p>(log⁡m)AU(m)μ2(m)αω(m)(∑l|mξ(dl))∑∑∗amodm⁡|K(ad,m)|×∑n≡amodm(n,md)=1nd>mV(nd)g(mndX)μ2(n)(2β)ω(n). This can be regarded as a kind of distributions of $$(2\beta)^{\omega(n)}$$ in arithmetic progressions on average. For each fixed $$d\leqslant X^\delta$$, we may appeal to Lemma 3.1, getting an acceptable approximation with an error term $$O(X(\log X)^{-100})$$. We know from Lemma 3.7 that   ∑∑∗amodm⁡|K(a,m)|=∏p|m∑∑∗amodp⁡|K(a,p)|=∏p|m(43πφ(p)+O(p1/2))=(43π)ω(m)φ(m)∏p|m(1+O(p−1/2)). Hence we may derive   H2(X,H)=∑l∑m≡0modlp|m⇒p>(log⁡m)AU(m)μ2(m)(8α3π)ω(m)×∑∑(nd,m)=1nd>m⁡V(nd)g(mndX)μ2(nd)(2β)ω(nd)ξ(dl)+o(Xlog⁡X)=∑∑p|m⇒p>(log⁡m)Am<n⁡U(m)V(n)g(mnX)μ2(mn)(8α3π)ω(m)(2β)ω(n)(∑d|mnλd)2+o(Xlog⁡X). Taking into account all admissible tuples $$\mathbf{H}$$, we find   H2(X)⩽∑ng(nX)μ2(n)τ(n;8α3π,2β)(∑d|nλd)2+O(δXlog⁡X). Here $$O(\delta{X}/\log X)$$ comes from those $$M,N$$ with $$M\leqslant N\leqslant X^{1/2+2\delta}$$ and $$MN=X$$. Note that $$\delta$$ is sufficiently small; this is acceptable in later numerical computations. Let $$k$$ be a positive integer and take   α=3π16k,   β=14k, so that   τ(n;8α3π,2β)=τ(n;k2,k2)⩽12kω(n) for squarefree $$n$$. Hence the above upper bound for $$H_2(X)$$ becomes   H2(X)⩽12∑ng(nX)μ2(n)kω(n)(∑d|nλd)2+O(δXlog⁡X). Proposition 2.2 then follows from the following proposition. Proposition 4.1. Let $$\kappa$$ be a fixed positive integer and $$F$$ vanish at $$0$$ to the order at least $$\kappa.$$ Under the choice (2.2) for $$(\lambda_d),$$ we have   ∑ng(nX)μ2(n)κω(n)(∑d|nλd)2=4g~(1)c(κ,F)Xlog⁡X(1+o(1)), where $$\mathfrak{c}(\kappa,F)$$ is defined by   c(k,F)=∑j=1k1Γ(j)2(kj)∫01F(j)(x)2(1−x)j−1(3+x)j−1dx. (4.6) □ 5 Proof of Proposition 4.1 We prove Proposition 4.1 in this section. Denote by $$\mathcal{N}(X)$$ the sum in question. Hence we have   N(X)=∑dξ(d)∑n≡0moddg(nX)μ2(n)κω(n)=∑dξ(d)κω(d)∑(n,d)=1g(ndX)μ2(n)κω(n), where $$\xi(d)$$ is defined by (4.5). By Mellin inversion, we can write   N(X)=12πi∫2−i∞2+i∞g~(s)(∑dξ(d)κω(d)ds)T(d,s)Xsds, where $$T(d,s)$$ is defined by the Dirichlet series   T(d,s)=∑n⩾1(n,d)=1μ2(n)κω(n)n−s,  ℜs>1. For $$\Re s>1,$$ we have   T(d,s)=∏p∤d(1+κps)=ζ(s)κS(s)∏p∣d(1+κps)−1, where   S(s)=∏p(1+κps)(1−1ps)κ. Hence we may write   N(X)=X2πi∫1−i∞1+i∞g~(s+1)A(s)ζ(s+1)κS(s+1)Xsds, (5.1) where   A(s)=∑dξ(d)κω(d)ds+1∏p∣d(1+κps+1)−1. 5.1 Analysis of $$\mathfrak{A}\boldsymbol{(s)}$$ Note that $$\mathfrak{A}(s)$$ is a finite sum over $$d$$ due to the support of $$\xi(d).$$ For $$s\in\mathbf{C}$$ and squarefree $$d\geqslant1$$, define   β(d,s)=ds+1κω(d)∏p∣d(1+κps+1). Therefore,   A(s)=∑d1∑d2λd1λd2β([d1,d2],s). For squarefree $$d_1,d_2,$$  1β([d1,d2],s)=1β(d1,s)β(d2,s)∏p∣(d1,d2)β(p,s)=1β(d1,s)β(d2,s)∑l∣(d1,d2)β∗(l,s), where $$\beta^*(l,s)$$ is defined by $$\beta^*(p,s)=\beta(p,s)-1.$$ Hence we have   A(s)=∑l⩽Dβ∗(l,s)Z(l,s)2 (5.2) with   Z(l,s)=∑d≡0modlλdβ(d,s). Recalling the definition (2.2), we have   Z(l,s)=∑d<Dd≡0modlμ(d)β(d,s)F(log⁡(D/d)log⁡D)=μ(l)β(l,s)∑d<D/l(d,l)=1μ(d)β(d,s)F(log⁡(D/(dl))log⁡D). By Mellin inversion, we may write   Z(l,s)=μ(l)β(l,s)12πi∫2−i∞2+i∞G(l;s,t)PˇD/l(t)(D/l)ttdt, where $$\check{P}_{\sqrt{D}/l}$$ is defined by (3.2) with   P(x)=F(x⋅log⁡(D/l)log⁡D), and   G(l;s,t)=∑d⩾1(d,l)=1μ(d)β(d,s)dt,   ℜ(s+t)>0, ℜs>−1. Note that   PˇD/l(t)=FˇD(t), we then have   Z(l,s)=μ(l)β(l,s)12πi∫2−i∞2+i∞G(l;s,t)FˇD(t)(D/l)ttdt, from which and (5.2) we derive that   A(s)=1(2πi)2∫2−i∞2+i∞∫2−i∞2+i∞FˇD(t1)FˇD(t2)(D)t1+t2∑l⩽Dμ2(l)β∗(l,s)β(l,s)2G(l;s,t1)G(l;s,t2)lt1+t2dt1dt2t1t2. Again by Mellin inversion, we conclude from (5.1) that   N(X)=X(2πi)4⨌g~(s+1)ζ(s+1)κS(s+1)FˇD(t1)FˇD(t2)H(s,t1,t2,w)×Xs(D)t1+t2+wt1t2wdsdt1dt2dw, where each integral is over $$2+it,t\in\mathbf{R},$$ and   H(s,t1,t2,w)=∑l⩾1μ2(l)β∗(l,s)β(l,s)2G(l;s,t1)G(l;s,t2)lt1+t2+w for $$\Re(s+t_1+t_2+w)>0,\Re(s+t_1)>0,\Re(s+t_2)>0$$, and $$\Re s>-1$$. Note that   H(s,t1,t2,w)=∏p(1−1β(p,s)pt1)(1−1β(p,s)pt2)×∑l⩾1μ2(l)β∗(l,s)β(l,s)2lt1+t2+w∏p∣l(1−1β(p,s)pt1)−1(1−1β(p,s)pt2)−1=∏p(1−1β(p,s)pt1)(1−1β(p,s)pt2)(1+β∗(p,s)pw(β(p,s)pt1−1)(β(p,s)pt2−1)). Since   β(p,s)=ps+1κ+1, we then have   H(s,t1,t2,w)=(ζ(s+t1+t2+w+1)ζ(s+t1+1)ζ(s+t2+1))κ×∏p(1−1β(p,s)pt1)(1−1ps+t1+1)−κ(1−1β(p,s)pt2)(1−1ps+t2+1)−κ×∏p(1+β∗(p,s)pw(β(p,s)pt1−1)(β(p,s)pt2−1))(1−1ps+t1+t2+w+1)κ. Put   K(s,t1,t2,w)=(ζ(s+1)ζ(s+t1+t2+w+1)ζ(s+t1+1)ζ(s+t2+1))κ(s(s+t1+t2+w)(s+t1)(s+t2))κ×∏p(1−1β(p,s)pt1)(1−1ps+t1+1)−κ(1−1β(p,s)pt2)(1−1ps+t2+1)−κ×∏p(1+β∗(p,s)pw(β(p,s)pt1−1)(β(p,s)pt2−1))(1−1ps+t1+t2+w+1)κ, it follows that   H2(X)=X(2πi)4⨌g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,w)×((s+t1)(s+t2)s(s+t1+t2+w))κXs(D)t1+t2+wt1t2wdsdt1dt2dw. 5.2 Shifting contours Now we are in the position to evaluate the multiple-integral by shifting contours. To this end, we define   C={−12016log⁡(|t|+2)+it:t∈R}, which is related to the zero-free region of Riemann zeta functions (see [2], for instance). We can shift all contours to $$\sigma=1/\log X$$ without passing any poles of the integrand. We now continue to shift the four contours to $$\mathcal{C}$$ one by one; we consider the $$w$$-integral first. There are two singularities $$w=0$$ and $$w=-(s+t_1+t_2)$$, which are of order 1 and $$k$$, respectively. Hence, after the shifting, the new integrand becomes   g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,0)((s+t1)(s+t2)s(s+t1+t2))κXs(D)t1+t2t1t2+g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,−(s+t1+t2))((s+t1)(s+t2)s)κXs(D)t1+t2t1t2×1(κ−1)!∂κ−1∂wκ−1(D)ww|w=−(s+t1+t2). In fact, there is also another contribution from the integral along $$\mathcal{C}$$, which is of a lower order of magnitude due to the growth of Riemann zeta functions (In the discussion below, we shall not present explicitly the error terms resulting from shifting contours). The second term comes from the singularity $$w=-(s+t_1+t_2),$$ and the factor $$(\sqrt{D})^{t_1+t_2}$$ will vanish after taking the partial derivatives, thus we conclude from Lemma 3.3 that the second term will produce a contribution of lower order of magnitude. We only consider the first term in latter discussions since what we are interested in is the constant in the main term. Now we are left with the triple-integral with respect to $$t_1,t_2$$, and $$s$$. The resultant integrand is   g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,0)((s+t1)(s+t2)s(s+t1+t2))κXs(D)t1+t2t1t2. We now shift the $$s$$-contour. Clearly, we shall encounter four singularities $$s=0,-t_1,-t_2$$, and $$-(t_1+t_2)$$. In fact, the latter three ones will produce factors of the shape $$(\sqrt{D}/X)^{t_1},(\sqrt{D}/X)^{t_2}$$, and $$(\sqrt{D}/X)^{t_1+t_2}$$. Following the same arguments as above, we conclude from Lemma 3.3 that all of these will contribute negligibly. Hence we need only consider the singularity $$s=0.$$ Note that   ((s+t1)(s+t2)s(s+t1+t2))κ=(1+t1t2s(s+t1+t2))κ=∑j=0k(κj)(t1t2s(s+t1+t2))j, thus we can rewrite the integrand as   ∑j=0κ(κj)Kj(s,t1,t2), where   Kj(s,t1,t2)=g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,0)(t1t2s(s+t1+t2))jXs(D)t1+t2t1t2. For $$j\geqslant1,$$ we have   Ress=0Kj(s,t1,t2)=(1+o(1))g~(1)S(1)Γ(j)FˇD(t1)FˇD(t2)K(0,t1,t2,0)(t1t2)j−1(D)t1+t2×(∂j−1∂sj−1Xs(s+t1+t2)j)s=0=(1+o(1))g~(1)S(1)Γ(j)FˇD(t1)FˇD(t2)K(0,t1,t2,0)(t1t2)j−1×∑i=0j−1(j−1i)(log⁡X)j−i−1(−1)iΓ(j+i)Γ(j)(D)t1+t2(t1+t2)j+i. After shifting the $$s$$-contour, we are left with the integrand   g~(1)S(1)∑j=1κ(κj)∑i=0j−1(j−1i)(log⁡X)j−i−1(−1)iΓ(j+i)Γ(j)2×FˇD(t1)FˇD(t2)K(0,t1,t2,0)(t1t2)j−1(D)t1+t2(t1+t2)j+i. Note that   K(0,0,0,0)=1S(1). As argued above, the expected main term comes from the singularity $$(t_1,t_2)=(0,0).$$ By Lemma A2 in the Appendix, we may evaluate the residue of integrand at $$(0,0),$$ getting   N(X)=(1+o(1))g~(1)Xlog⁡X∑j=1κ∑i=0j−1(−1)i4j−iΓ(j)2(κj)(j−1i)∫01F(j)(x)2(1−x)j+i−1dx=4g~(1)(1+o(1))Xlog⁡X∑j=1κ1Γ(j)2(κj)∫01F(j)(x)2(1−x)j−1(3+x)j−1dx as stated in Proposition 4.1. 6 Concluding Theorem 1.1: Numerical computations In order to obtain a positive lower bound for $$H^\pm(X),$$ it suffices to choose $$\rho$$ so that   ρH1(X)>2H2(X)+|ρH3(X)| for $$X$$ large enough. To do so, we would like to choose $$F$$, $$k,\alpha,\beta$$ and $$\rho$$ such that   ρ⋅∑2⩽i⩽52iAi(F)Ci>4⋅c(k,F). (6.1) Recall that $$\alpha=3\pi k/16,\beta=k/4$$. Clearly, we would like to employ optimizations such that the admissible $$\rho$$ can be as small as possible for a given tuple $$(\alpha,\beta)$$. Since there are much room for different choices of $$\alpha,\beta$$ and $$F$$, it is difficult to make full optimizations. As a heuristic treatment, we drop the contributions from terms with $$i=3,4,5$$ in (6.1), and consider the ratio   c(k,F)A2(F)∝1F(1)2∑j=1k1Γ(j)2(kj)∫01F(j)(x)2(1−x)j−1(3+x)j−1dx:=r(F). Take $$k=6$$, and put   F(x)=F(a,x)=x6(a0+a1x+a2x2+a3x3+a4x4), where $$\mathbf{a}=(a_0,a_1,a_2,a_3,a_4)\in\mathbf{R}^5$$ will be chosen later. After a routine calculation, $$r(F)$$ can be expressed as a quadratic form in $$\mathbf{a}$$. With the help of Mathematica 9, we may choose   a≈(42566991448102,−50062171448102,32207511448102,−37464494344308,67705734344308) to minimize the quadratic form. Upon such choice of $$F$$, (6.1) holds with   α=9π8,  β=32,  ρ=5.0×104. Hence the Kloosterman sum $$S(1,1;n)$$ changes sign for infinitely many $$n$$ with   τ(n;9π8,32)<5.0×104. (6.2) A crude inequality $$\omega(n)\leqslant10$$ then follows by observing that $$\tau(n;\frac{9\pi}{8},\frac{3}{2})\geqslant3^{\omega(n)}/2$$. In order to conclude Theorem 1.1, we would like explore a better control on $$\omega(n)$$ by appealing to a more careful analysis on the combinatorics of $$\tau(n;\frac{9\pi}{8},\frac{3}{2}).$$ Let $$X$$ be a sufficiently large number. Given a squarefree number $$n\in(X,2X],$$ we suppose $$p_1,p_2,\ldots,p_s$$ are the prime factors of $$n$$ if $$\omega(n)=s.$$ For each $$p_j,1\leqslant j\leqslant s$$, we set   pj=nθj,  0⩽θ1⩽θ2⩽⋯⩽θs, so that $$\theta_1+\theta_2+\cdots+\theta_s=1.$$ For $$\alpha,\beta\in\mathbf{R}_+,$$ we thus have   τ(n;α,β)=βsτ(θ1,θ2,…,θs) with   τ(θ1,θ2,…,θs)=∑0⩽j⩽s−1(αβ)j∑I⊂{1,2,…,s}|I|=j∑i∈Iθi⩽1/21. The minimum of $$\tau(\theta_1,\theta_2,\ldots,\theta_s)$$ is reached at $$\theta_1=\theta_2=\cdots=\theta_s=1/s$$ if $$\alpha\geqslant\beta.$$ In particular, for $$s=8$$, we have   τ(θ1,θ2,…,θ8)⩾∑0⩽j⩽4(αβ)j(8j). It follows that   τ(n;9π8,32)⩾(32)8∑0⩽j⩽4(3π4)j(8j)>7.8×104 if $$\omega(n)=8$$, which contradicts (6.2) unless $$\omega(n)\leqslant7$$. This completes the proof of Theorem 1.1. The Mathematica codes can be found at http://gr.xjtu.edu.cn/web/ping. or requested from the author. Funding This work was supported by PSF of Shaanxi Province, CPSF [No. 2015M580825], and NSF [No. 11601413] of People’s Republic of China. Acknowledgement The author would like to thank Étienne Fouvry and Philippe Michel for their valuable comments. Appendix. Computation of residues The Appendix is devoted to a calculation on double residues of certain meromorphic functions in two variables that appears in the proof of Proposition 4.1. We first state an auxiliary result that can be regarded as a special case. This can be proved following an approach of Motohashi as argued in [15, Lemma 6]. Lemma A1. Suppose $$M\geqslant1,$$ and $$k_1,k_2,l$$ are non-negative integers. Then we have   Res(s1,s2)=(0,0)Ms1+s2(s1+s2)ls1k1+1s2k2+1=1(k1+k2+l)!(k1+k2k1)(log⁡M)k1+k2+l. □ Lemma A2. Let $$v,v_1,v_2$$ be fixed positive integers. Suppose $$P,Q$$ are two smooth functions that have zeros of orders at least $$v_1,v_2$$ at $$0,$$ respectively, and $$W(t_1,t_2)$$ is holomorphic in the right half plane containing a neighborhood of $$(0,0)$$ with $$W(0,0)\neq0.$$ Put   R:=Res(t1,t2)=(0,0)PˇM(t1)QˇM(t2)W(t1,t2)t1v1−1t2v2−1Mt1+t2(t1+t2)v. Then we have   R=(1+o(1))W(0,0)(log⁡M)v−v1−v2(v−1)!∫01P(v1)(x)Q(v2)(x)(1−x)v−1dx, where $$P^{(v_1)}$$ denotes the $$v_1$$th derivatives of $$P$$ and similarly for $$Q^{(v_2)}.$$ □ Proof Recall that   PˇM(s)=∑k⩾0ak(slog⁡M)k,  QˇM(s)=∑k⩾0bk(slog⁡M)k. Thus,   R=∑k1⩾v1∑k2⩾v2ak1bk2(log⁡M)k1+k2Res(t1,t2)=(0,0)W(t1,t2)Mt1+t2t1k1−v1+1t2k2−v2+1(t1+t2)v. It follows from Lemma A1 that   R=(1+o(1))W(0,0)(log⁡M)v−v1−v2∑k1⩾v1∑k2⩾v2ak1bk2(k1+k2+v−v1−v2)!(k1+k2−v1−v2k1−v1)=(1+o(1))W(0,0)(log⁡M)v−v1−v2∫[v],1P(v1)(x)Q(v2)(x)dx, where $$\int_{[v],t}$$ is defined recursively by   ∫[1],tf(x)dx=∫0tf(x)dx,   ∫[v],tf(x)dx=∫0tdx∫[v−1],xf(y)dy. Put   rv(t)=∫[v],tf(x)dx,   v⩾1. Thus   ∂∂trv+1(t)=rv(t),   v⩾1. Consider the formal power series   R(y,t)=∑v⩾2rv(t)yv. Hence   ∂∂tR(y,t)=∑v⩾2rv−1(t)yv=∑v⩾1rv(t)yv+1=yR(y,t)+y2r1(t). Solving the differential equation with respect to $$t$$, we get   R(y,t)=y2eyt(∫0tr1(s)e−ysds+c), where $$c$$ is some constant. From integration by parts, we derive that   ∫0tr1(s)e−ysds=1y(−e−ytr1(t)+∫0tf(s)e−ysds), from which we conclude   R(y,t)=cy2eyt+yr1(t)−yeyt∫0tf(s)e−ysds. Comparing the coefficients of the expansions on both sides of the identity   ∑v⩾2rv(t)yv=cy2eyt−yr1(t)+yeyt∫0tf(s)e−ysds, we find   rv(t)=c(v−2)!tv−2+1(v−1)!∫0tf(s)(t−s)v−1ds,   v⩾2. We now determine the value of $$c$$. Taking $$v=2$$, we get   r2(t)=c+∫0tf(s)(t−s)ds=c+∫0t(t−s)dr1(s)=c+r2(t) from integration by parts, from which we conclude that $$c=0$$. Hence we arrive at the expression   rv(t)=1(v−1)!∫0tf(s)(t−s)v−1ds,   v⩾1. Finally, we obtain   R=(1+o(1))W(0,0)(log⁡M)v−v1−v2(v−1)!∫01P(v1)(x)Q(v2)(x)(1−x)v−1dx as expected. ■ Footnotes 1(One should remember the level here is different from the usual convention in sieve theory.) References [1] Bombieri E. Friedlander J. B. and Iwaniec. H. “Primes in arithmetic progressions to large moduli.” Acta Mathematica  156 ( 1986): 203– 51. Google Scholar CrossRef Search ADS   [2] de La Vallée Poussin C.-J. “Sur la fonction zêta de Riemann et le nombre des nombres premiers inférieurs a une limite donnée.” Mémoires couronnés de l’Académie de Belgique  59 ( 1899): 1– 74. [3] Deligne P. “La conjecture de Weil II.” Publications Mathématiques de l’IHÉS  52 ( 1980): 137– 252. Google Scholar CrossRef Search ADS   [4] Fouvry É. “Sur le problème des diviseurs de Titchmarsh.” Journal für die reine und angewandte Mathematik  357 ( 1985): 51– 76. [5] Fouvry É. and Michel. Ph. “Crible asymptotique et sommes de Kloosterman.” Proceedings of Session in Analytic Number Theory and Diophantine Equations , 27 pp. Bonner Mathematische Schriften, vol. 360. University of Bonn, Bonn, 2003. [6] Fouvry É. and Michel. Ph. “Sur le changement de signe des sommes de Kloosterman.” Annals of Mathematics  165 ( 2007): 675– 715. Google Scholar CrossRef Search ADS   [7] Fouvry É. Kowalski E. and Michel. Ph. “Algebraic trace functions over the primes.” Duke Mathematical Journal  163 ( 2014): 1683– 736. Google Scholar CrossRef Search ADS   [8] Fouvry É. Kowalski E. and Michel. Ph. “Trace Functions over Finite Fields and their Applications.” Colloquium de Giorgi 2013 and 2014 , vol. 5, Colloquia, 7– 35, Ed. Norm., Pisa, 2015. Google Scholar CrossRef Search ADS   [9] Katz N. M. Sommes Exponentielles . Asterisque 79. Société Mathématique de France, 1980. 209 pp. [10] Katz N. M. Gauss Sums, Kloosterman Sums, and Monodromy Groups . Annals of Mathematics Studies, vol. 116. Princeton, NJ: Princeton University Press, 1988. [11] Kowalski E. Ph. Michel and VanderKam. J. “Non-vanishing of high derivatives of automorphic $$L$$-functions at the center of the critical strip.” Journal für die reine und angewandte Mathematik  526 ( 2000): 1– 34. Google Scholar CrossRef Search ADS   [12] Matomäki K. “A note on signs of Kloosterman sums.” Bulletin de la Société Mathématique de France  139 ( 2011): 287– 95. Google Scholar CrossRef Search ADS   [13] Michel Ph. “Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman, I.” Inventiones mathematicae  121 ( 1995): 61– 78. Google Scholar CrossRef Search ADS   [14] Sivak-Fischler J. “Crible asymptotique et sommes de Kloosterman.” Bulletin de la Société Mathématique de France  137 ( 2009): 1– 62. Google Scholar CrossRef Search ADS   [15] Xi P. “Sign changes of Kloosterman sums with almost prime moduli.” Monatshefte für Mathematik  177 ( 2015): 141– 63. Google Scholar CrossRef Search ADS   Communicated by Prof. Valentin Blomer © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Mathematics Research Notices Oxford University Press

# Sign Changes of Kloosterman Sums with Almost Prime Moduli. II

, Volume 2018 (4) – Feb 1, 2018
28 pages

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Oxford University Press
ISSN
1073-7928
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1687-0247
D.O.I.
10.1093/imrn/rnw276
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### Abstract

Abstract We prove that the Kloosterman sum $$S(1,1;c)$$ changes sign infinitely often as $$c$$ runs over squarefree numbers with at most 7 prime factors, which improves the previous results of Fouvry and Michel, Sivak-Fischler, Matomäki, and the author. 1 Introduction We continue our study on sign changes of Kloosterman sums, which can be defined by   S(m,n;c)=∑∑∗amodc⁡e(ma+na¯c) for each positive integer $$c$$ and integers $$m,n$$, where $$a\overline{a}\equiv1\bmod c.$$ Let us state our theorem quickly. Theorem 1.1. There exists an absolute constant $$c_0>0$$ such that for sufficiently large $$X>0$$,   |{c⩽X:S(1,1;c)≷0, ω(c)⩽7, c squarefree}|⩾c0Xlog⁡X, where $$\omega(c)$$ denotes the number of distinct prime factors of $$c$$. □ In view of the Weil bound   |S(m,n;p)|⩽2p,  (m,n,p)=1 (1.1) for each prime $$p$$, the main motivation of Theorem 1.1 is the Sato–Tate conjecture of Katz [9], which asserts that the normalized Kloosterman sums $$S(1,1,p)/2\sqrt{p}$$ become equidistributed with respect to the Sato–Tate measure as $$p$$ runs over all the primes; it would then follow immediately that $$S(1,1;p)$$ changes sign infinitely often as $$p$$ varies. The study on sign changes was initiated by Fouvry and Michel [5, 6], who obtained a weaker constant 23 by a pioneering combination and application of the Selberg sieve, spectral theory of automorphic forms and $$\ell$$-adic cohomology. The subsequent improvements are due to Sivak-Fischler [14], Matomäki [12], and the author [15], where the relevant constants are 18, 15, and 10, respectively. As in previous works, the proof of Theorem 1.1 also utilizes the Selberg sieve, and the framework is based on that in [15]. The main ingredient in this paper is the introduction of a truncated divisor function and a successful application of Sato–Tate distributions of Kloosterman sums in the vertical direction. The framework of proof will be outlined in Section 2. 1.1 Notation and convention As usual, $$\mu,\varphi$$ denote the Möbius and Euler functions, respectively. The superscript $$*$$ in summation indicates primitive elements. We use $$\varepsilon$$ to denote a sufficiently small positive number and $$A$$ a reasonably large number, which are not to be the same on each occasion. For $$\alpha,\beta>0$$, define a truncated divisor function   τ(n;α,β)=∑dl=nd<lp|d⇒p>(logn)Aαω(d)βω(l). We also define   σ−1∗(n)=∑d|n,d>11d. Throughout this paper, we assume $$g(x)$$ is a fixed non-negative smooth function supported in $$[1,2]$$. and its Mellin transform is defined as   g~(s)=∫0+∞g(x)xs−1dx. Integrating by parts, we have   g~(s)≪(|s|+1)−A for any $$A\geqslant0.$$ Moreover, we input the normalization   g~(1)=∫0+∞g(x)dx=1. 2 Outline of the proof We prove Theorem 1.1 by applying the Selberg sieve. Let $$\lambda=(\lambda_d)$$ be the Selberg sieve weight of level $$\sqrt{D}:=X^{1/4}\exp(-\sqrt{\log X})$$. We would like to start from the following weighted sum   H±(X)=∑ng(nX)|S(1,1;n)|±S(1,1;n)nμ2(n){ρ−τ(n;α,β)}(∑d|nλd)2, (2.1) where $$\rho,\alpha,\beta>0$$ are three parameters to be chosen later.1 Upon suitable choices of $$\rho,\alpha,\beta$$, and $$(\lambda_d)$$, the positivity of $$H^\pm(X)$$ implies, for $$X$$ large enough, that there exists $$n\in(X,2X]$$ with   τ(n;α,β)<ρ for which $$S(1,1;n)\gtrless0$$. This will establish Theorem 1.1 from a careful analysis on the combinatorics of $$\tau(n;\alpha,\beta)$$. From (2.1), we have   H±(X)⩾ρH1(X)−2H2(X)±ρH3(X), where   H1(X)=∑ng(nX)|S(1,1;n)|nμ2(n)(∑d|nλd)2,H2(X)=∑ng(nX)|S(1,1;n)|nμ2(n)τ(n;α,β)(∑d|nλd)2,H3(X)=∑ng(nX)S(1,1;n)nμ2(n)(∑d|nλd)2. The lower bound for $$H^\pm(X)$$ follows from a lower bound for $$H_1(X)$$, an upper bound for $$H_2(X)$$, and a good control on the order of magnitude of $$H_3(X)$$. Let $$F(x)$$ be a fixed smooth function supported on $$[0,1]$$ and vanishes at 0 to a suitable order. We choose $$(\lambda_d)$$ such that   λd=μ(d)F(log⁡(D/d)log⁡D),  d⩽D. (2.2) The following lower bound for $$H_1(X)$$ was obtained in [12] and [15] for special choices of $$F$$. The general case can be treated similarly and we state the following proposition without proof. Proposition 2.1. For any sufficiently large $$X$$, we have   H1(X)⩾g~(1)∑2⩽i⩽52iAi(F)Ci⋅Xlog⁡X(1+o(1)), where $$C_2= 0.11109,\ C_3= 0.03557,\ C_4= 0.01184,\ C_5= 0.00396,$$  Ai(F)=∫⋯∫RiLi2(γ,F;X1−α2−⋯−αi,Xα2,…,Xαi)α2⋯αi(1−α2−⋯−αi)dα2⋯dαi,Li(F;α1,α2,…,αi)=∑A⊆{α1,α2,…,αi}∑α∈Aα<14(−1)|A|F(1−4∑α∈Aα), and   R2:={α2∈[η,1):(34+η)(1−α2)<α2<12},R3:={(α2,α3)∈[η,1)2:12(1−α2−α3)<α2,  α3<α2<1−α2−α3},R4:={(α2,α3,α4)∈[η,1)3:12(1−α2−α3−α4)<α2+α3}     ∩{(α2,α3,α4)∈[η,1)3:α4<α3<α2<1−α2−α3−α4},R5:={(α2,α3,α4,α5)∈[η,1)4:12(1−α2−α3−α4−α5)<α2+α3+α4}     ∩{(α2,α3,α4,α5)∈[η,1)4:12(α3+α4+α5)<α2}     ∩{(α2,α3,α4,α5)∈[η,1)4:α5<α4<α3<α2<1−α2−α3−α4−α5},η:=10−2016. Proposition 2.2. Let $$k$$ be a positive integer with $$\alpha=3\pi k/16,\beta=k/4$$, and $$\mathfrak{c}(k,F)$$ a constant defined by (4.6). For any sufficiently large $$X,$$ we have   H2(X)⩽2g~(1)⋅c(k,F)Xlog⁡X(1+o(1)). □ The proof of Proposition 2.2 will be presented in Section 4. The constant $$\frac{8}{3\pi}$$ comes from certain Sato–Tate distributions of Kloosterman sums in the vertical direction. The following proposition follows from the Bombieri–Vinogradov type estimate for Kloosterman sums that was initiated by Fouvry and Michel [6] deriving from the spectral theory of automorphic forms; the current version can be found in [15, Lemma 9]. Proposition 2.3. For any $$A>0$$ and sufficiently large $$X$$, we have   H3(X)≪X(log⁡X)−A, where the implied constant depends on $$A$$ and $$g$$. □ Theorem 1.1 follows from suitable choices of all parameters appearing in Propositions 2.1 and 2.2. The details, including the partial optimization on the choice of $$F$$, will be given in Section 6. 3 Auxiliary lemmas 3.1 Distribution in arithmetic progressions The following lemma is an extension of the classical Barban–Davenport–Halberstam theorem, which is originally devoted to primes in arithmetic progressions. The statement and proof can be found in [1, Theorem 0]. Lemma 3.1. Let $$(\alpha_n),n\leqslant N,$$ be any sequences satisfying the “Siegel–Walfisz” condition:   ∑n≡amodq(n,d)=1αn−1φ(q)∑(n,qd)=1αn≪τ(d)O(1)‖α‖N1/2(log⁡N)−C for any $$q\geqslant1,d\geqslant1,C>0,(q,a)=1,a\neq0$$ with an implied constant in $$\ll$$ depending only on $$C$$. Here $$\|\cdot\|$$ denotes the $$\ell_2$$-norm. Then, for any $$A>0,$$ there exists certain $$B=B(A)>0$$ such that   ∑q⩽Q∑∑∗amodq⁡|∑n≡amodqαn−1φ(q)∑(n,q)=1αn|2≪‖α‖2N(log⁡N)−A, provided that $$Q\leqslant N(\log N)^{-B},$$ where the implied constant depends only on $$A.$$ □ 3.2 Partition of unity We require the following lemma as a smooth partition of unity, which enables us to separate variables attaching smooth weights (see [4, Lemme 2], for instance). Lemma 3.2. For every $$\varDelta>1,$$ there exists a sequence $$\{b_{l,\varDelta}\}_{l\geqslant0}$$ of smooth functions with support included in $$[\varDelta^{l-1},\varDelta^{l+1}]$$, such that   ∑l⩾0bl,Δ(x)=1  for all x⩾1, and   xjbl,Δ(j)(x)≪(1−Δ−1)−j  for all x⩾1 and l⩾0. (3.1) □ 3.3 Mellin inversions We state two results from complex analysis as typical applications of the Mellin inversion formula. Lemma 3.3. Suppose $$k$$ is a non-negative integer. Then   k!2πi∫1−i∞1+i∞xssk+1ds={0,   0<x⩽1,(log⁡x)k,x>1, holds for any fixed positive number $$x.$$ □ Let $$P$$ be a smooth function with $$P(0)=0$$, and suppose it admits the following Taylor expansion   P(x)=∑k⩾0akk!xk. For $$N>1$$ and $$s\in\mathbf{C}\setminus\{0\}$$, define   PˇN(s)=∑k⩾0ak(slog⁡N)k. (3.2) Lemma 3.4. Suppose $$N>1$$ is not an integer. For any coefficients $$y_n$$ with $$y_n=O(\tau(n)^{O(1)}(\log n)^{O(1)}),$$ we have   ∑n⩽NynP(log⁡(N/n)log⁡N)=12πi∫2−i∞2+i∞PˇN(s)Y(s)Nssds, where   Y(s)=∑n⩾0ynns,   ℜs>1. □ Proof This is a consequence of [11, Lemma 2.1]. ■ 3.4 Sato–Tate distribution of Kloosterman sums From the works of Deligne [3] and Katz [10], it follows that the map   m↦S(m,1;p)p=2cos⁡θp(m),  m∈Fp× is the trace function of an $$\ell$$-adic sheaf $$\mathcal{K}l$$ on $$\mathbf{G}_{m}(\mathbf{F}_p)=\mathbf{F}_p^\times$$, which is of rank 2 and pure of weight 0. Therefore,   2cos⁡θp(m)=tr(Frobm,Kl). A celebrated theorem of Katz states that the set $$\{ \theta_p(m):m \in\mathbf{F}_p^\times\}$$ becomes equidistributed with respect to the Sato–Tate measure $$\mu_{\rm ST}=\frac{2}{\pi}\sin\theta\mathrm{d}\theta$$ as $$p \rightarrow \infty$$. By the Weyl equidistribution criterion and the Peter–Weyl theorem, the proof of Katz’s Sato–Tate distribution reduces to the study of   ∑m∈Fp×symk(θp(m))=∑m∈Fp×tr(Frobm,symkKl), where $$\mathrm{sym}^k\mathcal{K}l$$ is the $$k$$th symmetric power of the Kloosterman sheaf $$\mathcal{K}l$$ (i.e., the composition of the sheaf $$\mathcal{K}l$$ with the $$k$$th symmetric power representation of $$SL_2$$) and   symk(θ)=sin⁡(k+1)θsin⁡θ. Then the equidistribution of Katz can be formulated equivalently as   |∑m∈Fp×symk(θp(m))|⩽12(k+1)p; (3.3) we refer to Example 13.6 and the preceding theorem in [10] for more details. In fact, the square-root cancellation also holds if replacing $$\theta_p(m)$$ by $$\theta_p(f(m))$$ for any non-constant rational function $$f$$ of fixed degree over $$\mathbf{F}_p^\times,$$ in which case the independence of the upper bound on $$k$$ is polynomial of higher degree. In particular, it was proved by Michel [13], for each fixed $$k\geqslant1$$, that   ∑m∈Fp×symk(θp(m¯2))≪kO(1)p, (3.4) where the implied constant is absolute. The sheaf here should be $$\mathrm{sym}^k([-2]^*\mathcal{K}l)$$. In fact, such square-root cancellation follows from the Riemann Hypothesis of Deligne [3] on the quasi-orthogonality of trace functions of geometrically irreducible sheaves. One can also refer to a formulation of Fouvry, Kowalski and Michel that is more accessible to analytic applications, see the survey [8, Theorem 4.1], for instance. In particular, we may state the following square-root cancellations as consequences of the Riemann Hypothesis of Deligne. Lemma 3.5. Let $$\chi,\psi$$ be multiplicative and additive characters modulo $$p$$, respectively. For each fixed positive integer $$k,$$ we have   ∑∑∗mmodp⁡χ(m)symk(θp(m¯2))≪kO(1)p, (3.5) and   ∑∑∗mmodp⁡ψ(m)symk(θp(m¯2))≪kO(1)p. (3.6) Suppose $$(n_1n_2,p)=1$$ and $$n_1\not\equiv\pm n_2\bmod p.$$ For each fixed positive integers $$k_1,k_2,$$ we have   ∑∑∗mmodp⁡ψ(m)symk1(θp(n1m¯2))symk2(θp(n2m¯2))≪(k1k2)O(1)p. (3.7) The above implied constants are all absolute. □ For $$(n,c)=1$$, put   K(n,c)=S(n¯2,1;c)c. (3.8) It follows from the Chinese Remainder Theorem that $$K(n,rs)=K(nr,s)K(ns,r)$$ for all $$n,r,s$$ with $$(r,s)=(n,rs)=1.$$ We also have the following correlations of Kloosterman sums. Lemma 3.6. For all $$n_1,n_2,h$$ with $$(n_1n_2,p)=1,$$ we have   ∑∑∗amodp⁡|K(n1a,p)||K(n2a,p)|e(hap)≪p1/2(h,p)1/2, where the implied constants are absolute. □ Proof If $$n_1\equiv \pm n_2\bmod p$$, the lemma follows from (3.6) and the expansion   |x|2=14+14sym2(x). The case of $$n_1\not\equiv\pm n_2\bmod p$$ is a consequence of (3.7) and the expansion (3.9)   |x|=43π+∑k⩾1βksym2k(x) (3.9) with $$\beta_k=O(1/k!)$$ from the theory of Chebyshev polynomials. ■ The following first moment of Kloosterman sums follows from (3.4) and the expansion (3.9). Lemma 3.7. For sufficiently large prime $$p,$$ we have   ∑∑∗amodp⁡|K(a,p)|=83πp+O(p). □ We also require the following evaluation of bilinear sums of Kloosterman sums. Lemma 3.8. Let $$q$$ be a squarefree number and $$K$$ defined by (3.8). For any bounded real coefficients $$(a_m)$$ and $$(b_n)$$ with supports in $$(M,2M],(N,2N],$$ respectively, we have   ∑∑(mn,q)=1⁡ambn|K(mn,q)|=1φ(q)∑∑∗rmodq⁡|K(r,q)|∑∑(mn,q)=1⁡ambn+O(‖A‖‖B‖(MN)1/2(N−1/2qε+M−1/2q1/4+ε+3ω(q)σ−1∗(q)1/2)), where $$\|\cdot\|$$ denotes the $$\ell_2$$-norm and the implied constant depends only on $$\varepsilon.$$ □ Remark 1. The appearance of $$3^{\omega(q)}\sigma_{-1}^*(q)^{1/2}$$ in the error term shows that one cannot obtain the desired asymptotic evaluation if $$q$$ has small prime factors, in which case the advantage of bilinear forms becomes inapparent. While $$q$$ is a large prime, the Lemma 3.8 is an immediate consequence of the bilinear form estimates involving $$\mathrm{sym}_k(\theta_p(\overline{mn}^2))$$ following the approach of Pólya–Vinogradov method, and a relevant study (with slightly stronger estimates) can be found for instance in [13] and [7]. Nevertheless, Lemma 3.8 gives a typical asymptotic formula as long as $$N>p^{2\varepsilon}$$, $$M>p^{1/2+2\varepsilon}$$, and $$9^{\omega(q)}\sigma_{-1}^*(q)=o(1)$$. In particular, the last restriction holds on average if we assume $$q$$ has no prime factors below $$(\log q)^{2016}.$$ □ Proof We prove this lemma by the dispersion method. Put   Δ(A,B;q)=∑∑(mn,q)=1⁡ambn|K(mn,q)|−|A||B|φ(q)∑∑∗rmodq⁡|K(r,q)|, where   |A|=∑(m,q)=1am,   |B|=∑(n,q)=1bn. Thus   Δ(A,B;q)=∑∑∗rmodq⁡|K(r,q)|(∑∑mn≡rmodq⁡ambn−1φ(q)∑∑(mn,q)=1⁡ambn)=∑(m,q)=1αm∑∑∗rmodq⁡|K(r,q)|(∑n≡m¯rmodqbn−1φ(q)∑(n,q)=1bn). By Cauchy inequality, we have   Δ(A,B;q)2⩽‖A‖2Δ(B), (3.10) where   Δ(B)=∑(m,q)=1f(m){∑∑∗rmodq⁡|K(r,q)|(∑n≡m¯rmodqbn−1φ(q)∑(n,q)=1bn)}2. Here $$f$$ is a non-negative smooth function with compact support, containing $$(M,2M]$$, where $$f$$ takes value 1. Squaring out and switching summation, we may write   Δ(B)=Δ1(B)−2Δ2(B)+Δ3(B) with   Δ1(B)=∑∑(n1n2,q)=1⁡bn1bn2∑∑∗⁡∑∑∗r1,r2modqr1n2≡r2n1modq⁡|K(r1,q)||K(r2,q)|∑m≡n1¯r1modqf(m),  Δ2(B)=|B|φ(q)∑(n,q)=1bn∑∑∗⁡∑∑∗r1,r2modq⁡|K(r1,q)||K(r2,q)|∑m≡n¯r1modqf(m), and   Δ3(B)=|B|2φ(q)2(∑∑∗rmodq⁡|K(r,q)|)2∑(m,q)=1f(m). By Poisson summation,   Δ1(B)=1q∑h∈Zf^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗⁡∑∑∗r1,r2modqr1n2≡r2n1modq⁡|K(r1,q)||K(r2,q)|e(r1hn1¯q)=1q∑h∈Zf^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|e(rhq)=Δ11(B)+Δ12(B), where   Δ11(B)=f^(0)q∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|, and   Δ12(B)=1q∑h≠0f^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|e(rhq). From the orthogonality of Dirichlet characters, we derive that   Δ11(B)=f^(0)qφ(q)∑χmodq|∑nbnχ(n)|2|∑∑∗rmodq⁡χ(r)|K(r,q)||2=f^(0)qφ(q)∑dl=q∑∑∗χmodd⁡|∑(n,q)=1bnχ(n)|2|∑rmodq(r,q)=1χ(r)|K(r,q)||2, where $$\sideset{}{^*}\sum_\chi$$ denotes the summation over primitive characters $$\chi\bmod d$$. For $$dl=q$$ and $$(r,q)=1$$, we have $$K(r,q)=K(rd,l)K(rl,d)$$, which yields   ∑rmodq(r,q)=1χ(r)|K(r,q)|=χ¯(l)∑∑∗r1modd⁡χ(r1)|K(r1,d)|⋅∑∑∗r2modl⁡|K(r2,l)| for each primitive character $$\chi\bmod d.$$ The term with $$d=1$$ gives the contribution   f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2, which is expected to dominate the asymptotic evaluation of $$\Delta_{11}(\mathcal{B}).$$ The remaining contributions from $$d>1$$ are at most   ⩽f^(0)qφ(q)∑dl=qd>1l24ω(l)∑∑∗χmodd⁡|∑(n,q)=1bnχ(n)|2|∑∑∗rmodd⁡χ(r)|K(r,d)||2. The inner most sum over $$r$$ is at most $$O(d^{1/2}3^{\omega(d)})$$ in view of (3.5) and the Chinese Remainder Theorem. Thus the above quantity is bounded by   ≪f^(0)qφ(q)9ω(q)∑dl=qd>1dl2∑χmodd|∑(n,q)=1bnχ(n)|2≪‖B‖2M9ω(q)qφ(q)∑dl=qd>1dl2(d+N)≪‖B‖2Mqε+‖B‖2M N9ω(q)σ−1∗(q). We thus obtain that   Δ11(B)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O((qεM+MN9ω(q)σ−1∗(q))‖B‖2). From integration by parts, it follows that   f^(λ)≪M(1+M|λ|)−A for any fixed $$A\geqslant0$$. Thus,   Δ12(B)=1q∑|h|<q1+ε/Mf^(hq)∑∑(n1n2,q)=1⁡bn1bn2∑∑∗rmodq⁡|K(rn1,q)||K(rn2,q)|e(rhq)+O(1). In view of Lemma 3.6 and the Chinese Remainder Theorem, we derive that   Δ12(B)≪Mq∑|h|<q1+ε/M∑∑(n1n2,q)=1⁡|bn1|2q1/2+ε(h,q)1/2≪q1/2+εN‖B‖2. Therefore, we obtain   Δ1(B)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O(qε(M+q1/2N)‖B‖2). Similarly, we have   Δ2(B)=|B|qφ(q)∑h∈Zf^(hq)∑(n,q)=1bn∑∑∗⁡∑∑∗r1,r2modq⁡|K(r1,q)||K(r2,q)|e(r1hn¯q)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O(q1/2+εN‖B‖2), and   Δ3(B)=f^(0)|B|2qφ(q)(∑∑∗rmodq⁡|K(r,q)|)2+O(qεN‖B‖2). Collecting the above evaluations, we obtain   Δ(B)≪(q1/2+εN+qεM+MN9ω(q)σ−1∗(q))‖B‖2, from which and (3.10), Lemma 3.8 follows. ■ 4 Proof of Proposition 2.2: Upper bound for $$\boldsymbol{H}_{\boldsymbol{2}}\boldsymbol{(X)}$$ Recall the definition (3.8) of $$K(m,n)$$ and the twisted multiplicativity, then $$H_2(X)$$ becomes   H2(X)=∑ng(nX)|K(1,n)|μ2(n)τ(n;α,β)(∑d|nλd)2=∑∑m<np|m⇒p>(log⁡m)A⁡g(mnX)|K(m,n)||K(n,m)|μ2(mn)αω(m)βω(n)(∑d|mnλd)2. From the Weil bound for Kloosterman sums, it follows that   H2(X)⩽∑∑m<np|m⇒p>(log⁡m)A⁡g(mnX)|K(n,m)|μ2(mn)αω(m)(2β)ω(n)(∑d|mnλd)2. Following Lemma 3.2, we may separate variables $$m,n$$ by introducing a series of smooth functions of the shape $$b_{l,\varDelta}$$. To do so, we put   Δ=1+(log⁡X)−B for some parameter $$B\geqslant1.$$ Denote by $$M,N$$ some parameters of the form   M=Δl1=(1+(log⁡X)−B)l1,   N=Δl2=(1+(log⁡X)−B)l2 (4.1) for some $$l_1,l_2\in\mathbf{N}.$$ For the variables $$M=\varDelta^{l_1},N=\varDelta^{l_2}$$, we define   U=Ul1=bl1,Δ,   V=Vl2=bl2,Δ (4.2) as given in Lemma 3.2. Thus, the derivatives of $$U$$ and $$V$$ satisfy   xlU(l)(x),  xlV(l)(x)≪(log⁡X)Bl (4.3) for all $$l\geqslant 0$$, where the implied constant may depend on $$l$$. Moreover, the integration by parts gives   U^(ξ)≪M(log⁡X)Bl(1+|ξ|M)−l,   V^(ξ)≪N(log⁡X)Bl(1+|ξ|N)−l (4.4) for any fixed $$l\geqslant0$$, and the implied constant also depends on $$l$$. For $$\mathbf{H}=(M,N)$$, we consider the following sum   H2(X,H)=∑∑m<np|m⇒p>(log⁡m)A⁡U(m)V(n)g(mnX)|K(n,m)|μ2(mn)αω(m)(2β)ω(n)(∑d|mnλd)2, where $$U,V$$ are two smooth functions defined as above, so that   H2(X)⩽∑HH2(X,H). The above summation is restricted to at most $$O((\log X)^{2B})$$ tuples of $$\mathbf{H}=(M,N)$$ with $$M,N$$ given by (4.1). From the support of $$g$$ and the restriction that $$m<n$$, we may assume   MN≍X,   M≪N. Let $$\delta>0$$ be a sufficiently small number. For $$N>X^{1/2+2\delta}$$, we write   H2(X,H)=∑l∑m≡0modlp|m⇒p>(log⁡m)AU(m)μ2(m)αω(m)Ψ(N;m,l), where   ξ(n)=∑[d1,d2]=nλd1λd2, (4.5) and   Ψ(N;m,l):=∑∑(nd,m)=1nd>m⁡V(nd)g(mndX)|K(nd,m)|μ2(nd)(2β)ω(nd)ξ(dl). We evaluate $$H_2(X;M,N)$$ in two complementary methods. Note that $$dl\leqslant D\leqslant X^{1/2}$$ in view of the level of Selberg sieve weights. We divide the sum over $$d$$ to $$d\leqslant X^\delta$$ and $$d>X^\delta$$, so that $$\Psi(N;m,l)$$ can be split to   Ψ(N;m,l)=∑n∑d⩽Xδ+∑n∑d>Xδ=Ψ1(N;m,l)+Ψ2(N;m,l), say. From the support of $$V$$ we see $$nd\asymp N>X^{1/2+2\delta}.$$ Thus, $$n\gg X^{1/4+2\delta}$$ if $$d^\delta<d<X^{1/4+\delta}$$, and $$n\gg X^{2\delta}$$ if $$X^{1/4+\delta}\leqslant d<X^{1/2}$$. Furthermore, we have $$X^{1/4+2\delta}>M^{1/2+\delta}$$. It then follows from Lemma (3.8) that   Ψ2(N;m,l)=1φ(m)∑∑∗amodm⁡|K(a,m)|∑∑(nd,m)=1d>Xδ,nd>m⁡V(nd)g(mndX)μ2(nd)(2β)ω(nd)ξ(dl)+O(N1−δ3+(log⁡M)−A/4). If $$d\leqslant X^\delta$$, we see $$n\gg X^{1/2+\delta}$$ from the support of $$V$$. Note that $$m\asymp M<(N/d)^{1-\delta}.$$ In such case we may arrange the contributions, coming from $$\Psi_1(N;m,l)$$, in $$H_2(X;M,N)$$ as   =∑d<Xδμ2(d)(2β)ω(d)∑p|m⇒p>(log⁡m)AU(m)μ2(m)αω(m)(∑l|mξ(dl))∑∑∗amodm⁡|K(ad,m)|×∑n≡amodm(n,md)=1nd>mV(nd)g(mndX)μ2(n)(2β)ω(n). This can be regarded as a kind of distributions of $$(2\beta)^{\omega(n)}$$ in arithmetic progressions on average. For each fixed $$d\leqslant X^\delta$$, we may appeal to Lemma 3.1, getting an acceptable approximation with an error term $$O(X(\log X)^{-100})$$. We know from Lemma 3.7 that   ∑∑∗amodm⁡|K(a,m)|=∏p|m∑∑∗amodp⁡|K(a,p)|=∏p|m(43πφ(p)+O(p1/2))=(43π)ω(m)φ(m)∏p|m(1+O(p−1/2)). Hence we may derive   H2(X,H)=∑l∑m≡0modlp|m⇒p>(log⁡m)AU(m)μ2(m)(8α3π)ω(m)×∑∑(nd,m)=1nd>m⁡V(nd)g(mndX)μ2(nd)(2β)ω(nd)ξ(dl)+o(Xlog⁡X)=∑∑p|m⇒p>(log⁡m)Am<n⁡U(m)V(n)g(mnX)μ2(mn)(8α3π)ω(m)(2β)ω(n)(∑d|mnλd)2+o(Xlog⁡X). Taking into account all admissible tuples $$\mathbf{H}$$, we find   H2(X)⩽∑ng(nX)μ2(n)τ(n;8α3π,2β)(∑d|nλd)2+O(δXlog⁡X). Here $$O(\delta{X}/\log X)$$ comes from those $$M,N$$ with $$M\leqslant N\leqslant X^{1/2+2\delta}$$ and $$MN=X$$. Note that $$\delta$$ is sufficiently small; this is acceptable in later numerical computations. Let $$k$$ be a positive integer and take   α=3π16k,   β=14k, so that   τ(n;8α3π,2β)=τ(n;k2,k2)⩽12kω(n) for squarefree $$n$$. Hence the above upper bound for $$H_2(X)$$ becomes   H2(X)⩽12∑ng(nX)μ2(n)kω(n)(∑d|nλd)2+O(δXlog⁡X). Proposition 2.2 then follows from the following proposition. Proposition 4.1. Let $$\kappa$$ be a fixed positive integer and $$F$$ vanish at $$0$$ to the order at least $$\kappa.$$ Under the choice (2.2) for $$(\lambda_d),$$ we have   ∑ng(nX)μ2(n)κω(n)(∑d|nλd)2=4g~(1)c(κ,F)Xlog⁡X(1+o(1)), where $$\mathfrak{c}(\kappa,F)$$ is defined by   c(k,F)=∑j=1k1Γ(j)2(kj)∫01F(j)(x)2(1−x)j−1(3+x)j−1dx. (4.6) □ 5 Proof of Proposition 4.1 We prove Proposition 4.1 in this section. Denote by $$\mathcal{N}(X)$$ the sum in question. Hence we have   N(X)=∑dξ(d)∑n≡0moddg(nX)μ2(n)κω(n)=∑dξ(d)κω(d)∑(n,d)=1g(ndX)μ2(n)κω(n), where $$\xi(d)$$ is defined by (4.5). By Mellin inversion, we can write   N(X)=12πi∫2−i∞2+i∞g~(s)(∑dξ(d)κω(d)ds)T(d,s)Xsds, where $$T(d,s)$$ is defined by the Dirichlet series   T(d,s)=∑n⩾1(n,d)=1μ2(n)κω(n)n−s,  ℜs>1. For $$\Re s>1,$$ we have   T(d,s)=∏p∤d(1+κps)=ζ(s)κS(s)∏p∣d(1+κps)−1, where   S(s)=∏p(1+κps)(1−1ps)κ. Hence we may write   N(X)=X2πi∫1−i∞1+i∞g~(s+1)A(s)ζ(s+1)κS(s+1)Xsds, (5.1) where   A(s)=∑dξ(d)κω(d)ds+1∏p∣d(1+κps+1)−1. 5.1 Analysis of $$\mathfrak{A}\boldsymbol{(s)}$$ Note that $$\mathfrak{A}(s)$$ is a finite sum over $$d$$ due to the support of $$\xi(d).$$ For $$s\in\mathbf{C}$$ and squarefree $$d\geqslant1$$, define   β(d,s)=ds+1κω(d)∏p∣d(1+κps+1). Therefore,   A(s)=∑d1∑d2λd1λd2β([d1,d2],s). For squarefree $$d_1,d_2,$$  1β([d1,d2],s)=1β(d1,s)β(d2,s)∏p∣(d1,d2)β(p,s)=1β(d1,s)β(d2,s)∑l∣(d1,d2)β∗(l,s), where $$\beta^*(l,s)$$ is defined by $$\beta^*(p,s)=\beta(p,s)-1.$$ Hence we have   A(s)=∑l⩽Dβ∗(l,s)Z(l,s)2 (5.2) with   Z(l,s)=∑d≡0modlλdβ(d,s). Recalling the definition (2.2), we have   Z(l,s)=∑d<Dd≡0modlμ(d)β(d,s)F(log⁡(D/d)log⁡D)=μ(l)β(l,s)∑d<D/l(d,l)=1μ(d)β(d,s)F(log⁡(D/(dl))log⁡D). By Mellin inversion, we may write   Z(l,s)=μ(l)β(l,s)12πi∫2−i∞2+i∞G(l;s,t)PˇD/l(t)(D/l)ttdt, where $$\check{P}_{\sqrt{D}/l}$$ is defined by (3.2) with   P(x)=F(x⋅log⁡(D/l)log⁡D), and   G(l;s,t)=∑d⩾1(d,l)=1μ(d)β(d,s)dt,   ℜ(s+t)>0, ℜs>−1. Note that   PˇD/l(t)=FˇD(t), we then have   Z(l,s)=μ(l)β(l,s)12πi∫2−i∞2+i∞G(l;s,t)FˇD(t)(D/l)ttdt, from which and (5.2) we derive that   A(s)=1(2πi)2∫2−i∞2+i∞∫2−i∞2+i∞FˇD(t1)FˇD(t2)(D)t1+t2∑l⩽Dμ2(l)β∗(l,s)β(l,s)2G(l;s,t1)G(l;s,t2)lt1+t2dt1dt2t1t2. Again by Mellin inversion, we conclude from (5.1) that   N(X)=X(2πi)4⨌g~(s+1)ζ(s+1)κS(s+1)FˇD(t1)FˇD(t2)H(s,t1,t2,w)×Xs(D)t1+t2+wt1t2wdsdt1dt2dw, where each integral is over $$2+it,t\in\mathbf{R},$$ and   H(s,t1,t2,w)=∑l⩾1μ2(l)β∗(l,s)β(l,s)2G(l;s,t1)G(l;s,t2)lt1+t2+w for $$\Re(s+t_1+t_2+w)>0,\Re(s+t_1)>0,\Re(s+t_2)>0$$, and $$\Re s>-1$$. Note that   H(s,t1,t2,w)=∏p(1−1β(p,s)pt1)(1−1β(p,s)pt2)×∑l⩾1μ2(l)β∗(l,s)β(l,s)2lt1+t2+w∏p∣l(1−1β(p,s)pt1)−1(1−1β(p,s)pt2)−1=∏p(1−1β(p,s)pt1)(1−1β(p,s)pt2)(1+β∗(p,s)pw(β(p,s)pt1−1)(β(p,s)pt2−1)). Since   β(p,s)=ps+1κ+1, we then have   H(s,t1,t2,w)=(ζ(s+t1+t2+w+1)ζ(s+t1+1)ζ(s+t2+1))κ×∏p(1−1β(p,s)pt1)(1−1ps+t1+1)−κ(1−1β(p,s)pt2)(1−1ps+t2+1)−κ×∏p(1+β∗(p,s)pw(β(p,s)pt1−1)(β(p,s)pt2−1))(1−1ps+t1+t2+w+1)κ. Put   K(s,t1,t2,w)=(ζ(s+1)ζ(s+t1+t2+w+1)ζ(s+t1+1)ζ(s+t2+1))κ(s(s+t1+t2+w)(s+t1)(s+t2))κ×∏p(1−1β(p,s)pt1)(1−1ps+t1+1)−κ(1−1β(p,s)pt2)(1−1ps+t2+1)−κ×∏p(1+β∗(p,s)pw(β(p,s)pt1−1)(β(p,s)pt2−1))(1−1ps+t1+t2+w+1)κ, it follows that   H2(X)=X(2πi)4⨌g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,w)×((s+t1)(s+t2)s(s+t1+t2+w))κXs(D)t1+t2+wt1t2wdsdt1dt2dw. 5.2 Shifting contours Now we are in the position to evaluate the multiple-integral by shifting contours. To this end, we define   C={−12016log⁡(|t|+2)+it:t∈R}, which is related to the zero-free region of Riemann zeta functions (see [2], for instance). We can shift all contours to $$\sigma=1/\log X$$ without passing any poles of the integrand. We now continue to shift the four contours to $$\mathcal{C}$$ one by one; we consider the $$w$$-integral first. There are two singularities $$w=0$$ and $$w=-(s+t_1+t_2)$$, which are of order 1 and $$k$$, respectively. Hence, after the shifting, the new integrand becomes   g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,0)((s+t1)(s+t2)s(s+t1+t2))κXs(D)t1+t2t1t2+g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,−(s+t1+t2))((s+t1)(s+t2)s)κXs(D)t1+t2t1t2×1(κ−1)!∂κ−1∂wκ−1(D)ww|w=−(s+t1+t2). In fact, there is also another contribution from the integral along $$\mathcal{C}$$, which is of a lower order of magnitude due to the growth of Riemann zeta functions (In the discussion below, we shall not present explicitly the error terms resulting from shifting contours). The second term comes from the singularity $$w=-(s+t_1+t_2),$$ and the factor $$(\sqrt{D})^{t_1+t_2}$$ will vanish after taking the partial derivatives, thus we conclude from Lemma 3.3 that the second term will produce a contribution of lower order of magnitude. We only consider the first term in latter discussions since what we are interested in is the constant in the main term. Now we are left with the triple-integral with respect to $$t_1,t_2$$, and $$s$$. The resultant integrand is   g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,0)((s+t1)(s+t2)s(s+t1+t2))κXs(D)t1+t2t1t2. We now shift the $$s$$-contour. Clearly, we shall encounter four singularities $$s=0,-t_1,-t_2$$, and $$-(t_1+t_2)$$. In fact, the latter three ones will produce factors of the shape $$(\sqrt{D}/X)^{t_1},(\sqrt{D}/X)^{t_2}$$, and $$(\sqrt{D}/X)^{t_1+t_2}$$. Following the same arguments as above, we conclude from Lemma 3.3 that all of these will contribute negligibly. Hence we need only consider the singularity $$s=0.$$ Note that   ((s+t1)(s+t2)s(s+t1+t2))κ=(1+t1t2s(s+t1+t2))κ=∑j=0k(κj)(t1t2s(s+t1+t2))j, thus we can rewrite the integrand as   ∑j=0κ(κj)Kj(s,t1,t2), where   Kj(s,t1,t2)=g~(s+1)S(s+1)FˇD(t1)FˇD(t2)K(s,t1,t2,0)(t1t2s(s+t1+t2))jXs(D)t1+t2t1t2. For $$j\geqslant1,$$ we have   Ress=0Kj(s,t1,t2)=(1+o(1))g~(1)S(1)Γ(j)FˇD(t1)FˇD(t2)K(0,t1,t2,0)(t1t2)j−1(D)t1+t2×(∂j−1∂sj−1Xs(s+t1+t2)j)s=0=(1+o(1))g~(1)S(1)Γ(j)FˇD(t1)FˇD(t2)K(0,t1,t2,0)(t1t2)j−1×∑i=0j−1(j−1i)(log⁡X)j−i−1(−1)iΓ(j+i)Γ(j)(D)t1+t2(t1+t2)j+i. After shifting the $$s$$-contour, we are left with the integrand   g~(1)S(1)∑j=1κ(κj)∑i=0j−1(j−1i)(log⁡X)j−i−1(−1)iΓ(j+i)Γ(j)2×FˇD(t1)FˇD(t2)K(0,t1,t2,0)(t1t2)j−1(D)t1+t2(t1+t2)j+i. Note that   K(0,0,0,0)=1S(1). As argued above, the expected main term comes from the singularity $$(t_1,t_2)=(0,0).$$ By Lemma A2 in the Appendix, we may evaluate the residue of integrand at $$(0,0),$$ getting   N(X)=(1+o(1))g~(1)Xlog⁡X∑j=1κ∑i=0j−1(−1)i4j−iΓ(j)2(κj)(j−1i)∫01F(j)(x)2(1−x)j+i−1dx=4g~(1)(1+o(1))Xlog⁡X∑j=1κ1Γ(j)2(κj)∫01F(j)(x)2(1−x)j−1(3+x)j−1dx as stated in Proposition 4.1. 6 Concluding Theorem 1.1: Numerical computations In order to obtain a positive lower bound for $$H^\pm(X),$$ it suffices to choose $$\rho$$ so that   ρH1(X)>2H2(X)+|ρH3(X)| for $$X$$ large enough. To do so, we would like to choose $$F$$, $$k,\alpha,\beta$$ and $$\rho$$ such that   ρ⋅∑2⩽i⩽52iAi(F)Ci>4⋅c(k,F). (6.1) Recall that $$\alpha=3\pi k/16,\beta=k/4$$. Clearly, we would like to employ optimizations such that the admissible $$\rho$$ can be as small as possible for a given tuple $$(\alpha,\beta)$$. Since there are much room for different choices of $$\alpha,\beta$$ and $$F$$, it is difficult to make full optimizations. As a heuristic treatment, we drop the contributions from terms with $$i=3,4,5$$ in (6.1), and consider the ratio   c(k,F)A2(F)∝1F(1)2∑j=1k1Γ(j)2(kj)∫01F(j)(x)2(1−x)j−1(3+x)j−1dx:=r(F). Take $$k=6$$, and put   F(x)=F(a,x)=x6(a0+a1x+a2x2+a3x3+a4x4), where $$\mathbf{a}=(a_0,a_1,a_2,a_3,a_4)\in\mathbf{R}^5$$ will be chosen later. After a routine calculation, $$r(F)$$ can be expressed as a quadratic form in $$\mathbf{a}$$. With the help of Mathematica 9, we may choose   a≈(42566991448102,−50062171448102,32207511448102,−37464494344308,67705734344308) to minimize the quadratic form. Upon such choice of $$F$$, (6.1) holds with   α=9π8,  β=32,  ρ=5.0×104. Hence the Kloosterman sum $$S(1,1;n)$$ changes sign for infinitely many $$n$$ with   τ(n;9π8,32)<5.0×104. (6.2) A crude inequality $$\omega(n)\leqslant10$$ then follows by observing that $$\tau(n;\frac{9\pi}{8},\frac{3}{2})\geqslant3^{\omega(n)}/2$$. In order to conclude Theorem 1.1, we would like explore a better control on $$\omega(n)$$ by appealing to a more careful analysis on the combinatorics of $$\tau(n;\frac{9\pi}{8},\frac{3}{2}).$$ Let $$X$$ be a sufficiently large number. Given a squarefree number $$n\in(X,2X],$$ we suppose $$p_1,p_2,\ldots,p_s$$ are the prime factors of $$n$$ if $$\omega(n)=s.$$ For each $$p_j,1\leqslant j\leqslant s$$, we set   pj=nθj,  0⩽θ1⩽θ2⩽⋯⩽θs, so that $$\theta_1+\theta_2+\cdots+\theta_s=1.$$ For $$\alpha,\beta\in\mathbf{R}_+,$$ we thus have   τ(n;α,β)=βsτ(θ1,θ2,…,θs) with   τ(θ1,θ2,…,θs)=∑0⩽j⩽s−1(αβ)j∑I⊂{1,2,…,s}|I|=j∑i∈Iθi⩽1/21. The minimum of $$\tau(\theta_1,\theta_2,\ldots,\theta_s)$$ is reached at $$\theta_1=\theta_2=\cdots=\theta_s=1/s$$ if $$\alpha\geqslant\beta.$$ In particular, for $$s=8$$, we have   τ(θ1,θ2,…,θ8)⩾∑0⩽j⩽4(αβ)j(8j). It follows that   τ(n;9π8,32)⩾(32)8∑0⩽j⩽4(3π4)j(8j)>7.8×104 if $$\omega(n)=8$$, which contradicts (6.2) unless $$\omega(n)\leqslant7$$. This completes the proof of Theorem 1.1. The Mathematica codes can be found at http://gr.xjtu.edu.cn/web/ping. or requested from the author. Funding This work was supported by PSF of Shaanxi Province, CPSF [No. 2015M580825], and NSF [No. 11601413] of People’s Republic of China. Acknowledgement The author would like to thank Étienne Fouvry and Philippe Michel for their valuable comments. Appendix. Computation of residues The Appendix is devoted to a calculation on double residues of certain meromorphic functions in two variables that appears in the proof of Proposition 4.1. We first state an auxiliary result that can be regarded as a special case. This can be proved following an approach of Motohashi as argued in [15, Lemma 6]. Lemma A1. Suppose $$M\geqslant1,$$ and $$k_1,k_2,l$$ are non-negative integers. Then we have   Res(s1,s2)=(0,0)Ms1+s2(s1+s2)ls1k1+1s2k2+1=1(k1+k2+l)!(k1+k2k1)(log⁡M)k1+k2+l. □ Lemma A2. Let $$v,v_1,v_2$$ be fixed positive integers. Suppose $$P,Q$$ are two smooth functions that have zeros of orders at least $$v_1,v_2$$ at $$0,$$ respectively, and $$W(t_1,t_2)$$ is holomorphic in the right half plane containing a neighborhood of $$(0,0)$$ with $$W(0,0)\neq0.$$ Put   R:=Res(t1,t2)=(0,0)PˇM(t1)QˇM(t2)W(t1,t2)t1v1−1t2v2−1Mt1+t2(t1+t2)v. Then we have   R=(1+o(1))W(0,0)(log⁡M)v−v1−v2(v−1)!∫01P(v1)(x)Q(v2)(x)(1−x)v−1dx, where $$P^{(v_1)}$$ denotes the $$v_1$$th derivatives of $$P$$ and similarly for $$Q^{(v_2)}.$$ □ Proof Recall that   PˇM(s)=∑k⩾0ak(slog⁡M)k,  QˇM(s)=∑k⩾0bk(slog⁡M)k. Thus,   R=∑k1⩾v1∑k2⩾v2ak1bk2(log⁡M)k1+k2Res(t1,t2)=(0,0)W(t1,t2)Mt1+t2t1k1−v1+1t2k2−v2+1(t1+t2)v. It follows from Lemma A1 that   R=(1+o(1))W(0,0)(log⁡M)v−v1−v2∑k1⩾v1∑k2⩾v2ak1bk2(k1+k2+v−v1−v2)!(k1+k2−v1−v2k1−v1)=(1+o(1))W(0,0)(log⁡M)v−v1−v2∫[v],1P(v1)(x)Q(v2)(x)dx, where $$\int_{[v],t}$$ is defined recursively by   ∫[1],tf(x)dx=∫0tf(x)dx,   ∫[v],tf(x)dx=∫0tdx∫[v−1],xf(y)dy. Put   rv(t)=∫[v],tf(x)dx,   v⩾1. Thus   ∂∂trv+1(t)=rv(t),   v⩾1. Consider the formal power series   R(y,t)=∑v⩾2rv(t)yv. Hence   ∂∂tR(y,t)=∑v⩾2rv−1(t)yv=∑v⩾1rv(t)yv+1=yR(y,t)+y2r1(t). Solving the differential equation with respect to $$t$$, we get   R(y,t)=y2eyt(∫0tr1(s)e−ysds+c), where $$c$$ is some constant. From integration by parts, we derive that   ∫0tr1(s)e−ysds=1y(−e−ytr1(t)+∫0tf(s)e−ysds), from which we conclude   R(y,t)=cy2eyt+yr1(t)−yeyt∫0tf(s)e−ysds. Comparing the coefficients of the expansions on both sides of the identity   ∑v⩾2rv(t)yv=cy2eyt−yr1(t)+yeyt∫0tf(s)e−ysds, we find   rv(t)=c(v−2)!tv−2+1(v−1)!∫0tf(s)(t−s)v−1ds,   v⩾2. We now determine the value of $$c$$. Taking $$v=2$$, we get   r2(t)=c+∫0tf(s)(t−s)ds=c+∫0t(t−s)dr1(s)=c+r2(t) from integration by parts, from which we conclude that $$c=0$$. Hence we arrive at the expression   rv(t)=1(v−1)!∫0tf(s)(t−s)v−1ds,   v⩾1. Finally, we obtain   R=(1+o(1))W(0,0)(log⁡M)v−v1−v2(v−1)!∫01P(v1)(x)Q(v2)(x)(1−x)v−1dx as expected. ■ Footnotes 1(One should remember the level here is different from the usual convention in sieve theory.) References [1] Bombieri E. Friedlander J. B. and Iwaniec. H. “Primes in arithmetic progressions to large moduli.” Acta Mathematica  156 ( 1986): 203– 51. Google Scholar CrossRef Search ADS   [2] de La Vallée Poussin C.-J. “Sur la fonction zêta de Riemann et le nombre des nombres premiers inférieurs a une limite donnée.” Mémoires couronnés de l’Académie de Belgique  59 ( 1899): 1– 74. [3] Deligne P. “La conjecture de Weil II.” Publications Mathématiques de l’IHÉS  52 ( 1980): 137– 252. Google Scholar CrossRef Search ADS   [4] Fouvry É. “Sur le problème des diviseurs de Titchmarsh.” Journal für die reine und angewandte Mathematik  357 ( 1985): 51– 76. [5] Fouvry É. and Michel. Ph. “Crible asymptotique et sommes de Kloosterman.” Proceedings of Session in Analytic Number Theory and Diophantine Equations , 27 pp. Bonner Mathematische Schriften, vol. 360. University of Bonn, Bonn, 2003. [6] Fouvry É. and Michel. Ph. “Sur le changement de signe des sommes de Kloosterman.” Annals of Mathematics  165 ( 2007): 675– 715. Google Scholar CrossRef Search ADS   [7] Fouvry É. Kowalski E. and Michel. Ph. “Algebraic trace functions over the primes.” Duke Mathematical Journal  163 ( 2014): 1683– 736. Google Scholar CrossRef Search ADS   [8] Fouvry É. Kowalski E. and Michel. Ph. “Trace Functions over Finite Fields and their Applications.” Colloquium de Giorgi 2013 and 2014 , vol. 5, Colloquia, 7– 35, Ed. Norm., Pisa, 2015. Google Scholar CrossRef Search ADS   [9] Katz N. M. Sommes Exponentielles . Asterisque 79. Société Mathématique de France, 1980. 209 pp. [10] Katz N. M. Gauss Sums, Kloosterman Sums, and Monodromy Groups . Annals of Mathematics Studies, vol. 116. Princeton, NJ: Princeton University Press, 1988. [11] Kowalski E. Ph. Michel and VanderKam. J. “Non-vanishing of high derivatives of automorphic $$L$$-functions at the center of the critical strip.” Journal für die reine und angewandte Mathematik  526 ( 2000): 1– 34. Google Scholar CrossRef Search ADS   [12] Matomäki K. “A note on signs of Kloosterman sums.” Bulletin de la Société Mathématique de France  139 ( 2011): 287– 95. Google Scholar CrossRef Search ADS   [13] Michel Ph. “Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman, I.” Inventiones mathematicae  121 ( 1995): 61– 78. Google Scholar CrossRef Search ADS   [14] Sivak-Fischler J. “Crible asymptotique et sommes de Kloosterman.” Bulletin de la Société Mathématique de France  137 ( 2009): 1– 62. Google Scholar CrossRef Search ADS   [15] Xi P. “Sign changes of Kloosterman sums with almost prime moduli.” Monatshefte für Mathematik  177 ( 2015): 141– 63. Google Scholar CrossRef Search ADS   Communicated by Prof. Valentin Blomer © The Author 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

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