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The Quarterly Journal of Mathematics
, Volume Advance Article – Mar 2, 2018

8 pages

/lp/ou_press/on-the-pairwise-maxima-of-generalised-divisor-functions-qT2tv3mYec

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hay008
- Publisher site
- See Article on Publisher Site

Abstract In this paper, we prove the asymptotic growth rate of the summatory function of the pairwise maxima of the generalized divisor function dk(n), for a fixed positive integer k≥2. This result generalizes previous results of Kátai, Erdős and Hall on the local behaviour of divisor function on short intervals. 1. Introduction Let n be a natural number and let d(n) denote the number of divisors of n. Kátai, in his paper [3], studied the local behaviour of the function d(n). In his paper, he proved that ∑n≤xmax{d(n),d(n+1)}=2xlogx+O(x(logx)1−δ), (1.1)where δ is a suitable positive constant. In their paper [2], Erdős and Hall determined the following asymptotic for the local maxima of d(n): Theorem 1.1 (Erdős-Hall) If h=o((logx)3−22), then ∑n≤xmax{d(n),d(n+1),…,d(n+h−1)}=hxlogx+O(h2x(logx)2(2−1)). (1.2) In the case h=2, Equation (1.2) reduces to ∑n≤xmax{d(n),d(n+1)}=2xlogx+O(x(logx)2(2−1)). (1.3)Although the authors do not state this explicitly, with slight modifications their proof of Theorem 1.1 also provides us with ∑n≤xmax{d(n),d(n+h)}=2xlogx+O(x(logx)2(2−1)) (1.4)for fixed values of h. In this paper, we generalize (1.4) for fixed values of h and k by considering the relation ∑n≤xmax{dk(n),dk(n+h)}=∑n≤xdk(n)+∑n≤xdk(n+h)−∑n≤xmin{dk(n),dk(n+h)}=2∑n≤xdk(n)−∑n≤xmin{dk(n),dk(n+h)}+∑x<n≤x+hdk(n)−∑n<hdk(n)=2∑n≤xdk(n)+Ek(x,h). (1.5)Our main result is Theorem 1.2 below, which is proved in Section 3. Theorem 1.2 If hand kare fixed, then Ek(x,h)≪h,kx(logx)2(k−1) (1.6)as x→∞. By using the well-known asymptotic formula for the summatory function of dk(n) [6, p. 263], Theorem 1.2 states that if k>4 and h a fixed number, then ∑n≤xmax{dk(n),dk(n+h)}=2(k−1)!x(logx)k−1+O(x(logx)k−2) (1.7)and for k≤4 we have that ∑n≤xmax{dk(n),dk(n+h)}=2(k−1)!x(logx)k−1+O(x(logx)2(k−1)) (1.8)as x→∞. The main difficulty is that the approach of Erdős and Hall [2] breaks down for dk(n) if k≥4. Therefore, new ideas are necessary to generalize their results. To overcome such intricacies, we use a theorem by Nair and Tenenbaum [4] to obtain a bound on certain averages involving dk(n) which turns out to be sufficient to establish the asymptotic formula above. In Section 2 of the paper, we discuss the method of Erdős and Hall and why it breaks down when we try to generalize to dk(n). In Section 3, we prove Theorem 1.2, which is the main result of this paper. 2. The method of Erdős and Hall In this section, we briefly describe the method of proof of (1.4) used in their paper [2], and how it must be modified to establish Theorem 1.2. Note that d(pα)≥d(pα−1) for α≥1. Since d(n) is multiplicative, we have d(n)=∑d∣nf(d), (2.1)where f(pα)=g(pα)−g(pα−1)≥0 (2.2)for α≥1 and f(1)=1. The method of Erdős and Hall begins by using the simple facts that min{d(n),d(n+1)}≤d(n)d(n+1) (2.3)and ∑n≤xd(n)d(n+1)=∑n≤x∑d∣nf(d)∑e∣n+1f(e), (2.4)and a crucial step of their proof establishes that there exists a constant C such that d(n)=∑d∣nf(d)≤C∑d∣nd<nf(d). (2.5)To establish (2.5), the authors observe that ∑d∣nd≥nf(d)≤2logn∑d∣nd≥nf(d)logd≤2logn∑d∣nf(d)logd (2.6)for any multiplicative function f satisfying f(1)=1, so to prove (2.5), it is sufficient to establish the existence of a C′<1/2 such that ∑d∣nf(d)logd≤C′logn∑d∣nf(d) (2.7)because by (2.6), we then have ∑d∣nf(d)≤11−2C′∑d∣nd<nf(d). (2.8)However, we can prove that Lemma 2.1 For a multiplicative function fsatisfying f(1)=1, let g(n)=∑d∣nf(d), (2.9)then there exists a constant C′<1/2such that ∑d∣nf(d)logd≤C′logn∑d∣nf(d) (2.10)if and only if there exists a constant C″>1/2such that g(pα)≤1C″α∑j=0α−1g(pj) (2.11)for every pand every α≥1. Proof By logarithmic differentiation of ∑d∣nf(d)ds (2.12)one finds that ∑d∣nf(d)logd∑d∣nf(d)=∑pα∥n(f(p)+2f(p2)+⋯+αf(pα)1+f(p)+f(p2)+⋯+f(pα))logp. (2.13)From (2.13), it follows that the existence of C′ in (2.10) is equivalent to ∑j=0αjf(pj)≤C′α∑j=0αf(pj) (2.14)for every p and every α≥1. By (2.2) and some elementary analysis, (2.14) reduces to (2.11).□ Erdős and Hall prove that (2.11) holds when g(n)=d(n) so Lemma 2.1 applies. This gives a non-trivial estimate of (2.4) which implies Theorem 1.1. However, the following dilemma arises. Corollary 2.2 The growth constraint (2.11) does not hold for g(n)=dk(n)when k>3. Proof Since dk(pj)=(j+k−1j), we observe that (74)>12∑j=03(3+j3), (2.15)so (2.11) fails for g(n)=d4(n). Similar arguments show that (2.11) fails to hold for any k≥4.□ It follows from the previous corollary that Erdős and Hall approach does not apply for dk(n) for k≥4. We will remedy this in the next section. 3. A proof via the theorems of Nair–Tenenbaum and Selberg–Delange In this section, Theorem 1.2 is proved by establishing a suitable bound for the l.h.s. of (2.4) via Theorem 3.1 below, which is special case of a very general theorem of Nair and Tenenbaum [4] (Theorem 1 therein). Let Ω(n) denote the number of prime factors of n counted with multiplicity and let A and B be positive constants. Also let α>0 and ϵ>0 be quantities which may be taken to be arbitrarily small. Theorem 3.1 (Nair–Tenenbaum) If F1, F2are non-negative arithmetic functions satisfying F1(m)F2(n)≤min{AΩ(mn),B(ϵ)(mn)ϵ} (3.1)whenever (m,n)=1, then ∑x≤n≤x+yF1(n)F2(n+h)≪A,B,h,ϵy(logx)2∑mn≤xF1(m)F2(n)mn (3.2)uniformly for xα≤y≤x. From (2.3) and the fact that for fixed h, the sum ∑x<n≤x+hdk(n)≪h,kmaxn≤x+hdk(n)≪h,kkClog(x+h)/loglog(x+h)≪h,kxo(logk), (3.3)it follows from (1.5) that to prove Theorem 1.2, it will be sufficient to prove the following proposition. Proposition 3.2 For fixed hand kwe have ∑n≤xdk(n)dk(n+h)=O(x(logx)2(k−1)) (3.4)as x→∞. Proof Take F1(n)=F2(n)=dk(n) in Theorem 3.1, so that F1(m)F2(n)=dk(mn) when (m,n)=1. To begin, we must verify that (3.1) holds in this case, that is that dk(n)≤min{AΩ(n),B(ϵ)nϵ} (3.5)when n is squarefree. Since dk(p)=k it follows that dk(n)=kΩ(n), so we have A=k. Since Ω(n)=O(logn/loglogn) as n→∞ it follows that kΩ(n)≤B(ϵ)nϵ for every ϵ>0, so (3.5) holds in this case. For σ>1 let Dk(s)=∑1∞dk1/2(n)ns. (3.6)By the quantitative version of Perron’s formula—a general proof of which is given in Titchmarsh [6] (Lemma 3.12)—one now observes that for δ>0, k≥2, T>0 and x not an integer. We have ∑mn≤xF1(m)F2(n)mn=∑mn≤xdk1/2(m)dk1/2(n)mn=12πi∫δ−iTδ+iTDk2(s+1)xsdss+O(xδTDk2(δ+1))+O(logxTmaxn≤2x1n∑d∣ndk1/2(d)). (3.7)The remaining steps of the proof essentially follow the methods of Selberg [5] and Delange [1], which enable the integral on the r.h.s. of (3.7) to be estimated. This proceeds by evaluating the integral along segments marginally above and below the potential branch cut (−∞,0] and using Hankel’s integral representation of Γ(s). The first step is to observe that Dk2(s)=Hk(s)ζ2k1/2(s), (3.8)where Hk(s) has an absolutely convergent Euler product on compact subsets of the half plane σ>1/2. As such, for fixed k, ∣Hk(s)∣ is bounded above and away from zero on compact subsets of the half plane σ>1/2. Moreover, due to the simple pole of ζ(s) at s=1, from (3.8), it is evident that (−∞,0] is a branch cut for Dk2(s+1) whenever k is not square. Given ϵ>0, one takes the path of integration in (3.7) to consist of horizontal segments from δ−iT to −δ−iT and −δ+iT to δ+iT, vertical segments from −δ−iT to −δ−iϵ and −δ+iϵ to −δ+iT, and a truncated Hankel contour (a path from −δ−iϵ to −δ+iϵ passing around the cut along the segment [−δ,0], but not crossing it). From (3.8), the bounds on ∣Hk(s)∣ and the elementary fact that ζ(σ+it)=O(t1−σ+δ) for σ≥0, it is immediate that the vertical segments of the integral are ∣12πi∫−δ+iϵ−δ+iTHk(s+1)ζ2k1/2(s+1)xsdss∣≪k,δx−δT4δk1/2, (3.9)and that the horizontal segments of the integral are ∣12πi∫−δ+iTδ+iTHk(s+1)ζ2k1/2(s+1)xsdss∣≪k,δxδT4δk1/2−1. (3.10)Taking T=x2δ and δ=k−1/2/8, the r.h.s. of (3.9) is x−δ(x2δ)4δk1/2=x−δ+8δ2k1/2=x−δ+k−1/2/8=1 (3.11)and the r.h.s. of (3.10) is xδ(x2δ)4δk1/2−1=x−δ+8δ2k1/2=1, (3.12)so (3.9) and (3.10) are bounded as x→∞ for fixed k. Moreover, with these choices for δ and T, the first error term on the r.h.s. of (3.7) is xδTDk2(δ+1)=x−δDk2(δ+1)≪kx−δ (3.13)which is bounded as x→∞ for fixed k. The second error term on the r.h.s. of (3.7) is logxTmaxn≤2x1n∑d∣ndk1/2(d)≪kx−2δlogx(k+1)Clogx/loglogx≪kx−2δ+Clogk/loglogx, (3.14)which is also bounded as x→∞ for fixed k. For fixed k then, it follows that ∑mn≤xdk1/2(m)dk1/2(n)mn=12πi∫H(k,ϵ)Dk2(s+1)xsdss+Ok(1), (3.15)where the path of integration H(k,ϵ) is from −k−1/2/8−iϵ to −k−1/2/8+iϵ and not intersecting the half line (−∞,0]. Invoking (3.8) and the fact that ζ(s) has a simple pole at s=1, one may expand Hk(s+1) in a power series about s=0 to give Dk2(s+1)=∑n≤2k1/2cnsn−2k1/2+Ok(1) (3.16)so the r.h.s. of (3.15) is ∑n≤2k1/2cn2πi∫H(k,ϵ)xssn−2k1/2−1ds+Ok(1). (3.17)Making the change of variable s=z/logx in (3.17) then gives ∑n≤2k1/2cn(logx)2k1/2−n2πi∫H(k,ϵ,x)ezzn−2k1/2−1dz+Ok(1), (3.18)where H(k,ϵ,x) indicates a path of integration from −k−1/2logx/8−iϵlogx to −k−1/2logx/8+iϵlogx and not intersecting the half line (−∞,0]. Taking ϵ=o(1/logx), the path H(k,ϵ,x) approaches a standard Hankel contour H as x→∞ therefore, using Hankel’s identity 1Γ(s+1)=12πi∫Hezz−s−1dz, (3.19)in (3.18), from (3.7) we now have ∑mn≤xdk1/2(m)dk1/2(n)mn=∑n≤2k1/2cn(logx)2k1/2−nΓ(2k1/2−n+1)+Ok(1)=Ok((logx)2k1/2). (3.20)Thus, (3.19) and (3.2) together give ∑x≤n≤x+ydk1/2(n)dk1/2(n+h)≪h,ky(logx)2(k1/2−1) (3.21)uniformly for xα≤y≤x. To complete the proof of Proposition 3.2 we take y=x=2−m−1X successively in (3.20) and sum over the range 0≤m≤log2X, which gives ∑n≤Xdk1/2(n)dk1/2(n+h)X(logX)2(k1/2−1)≪h,k∑0≤m≤log2X2−m−1(1−(m−1)log2logX)2(k1/2−1)≪h,k1 (3.22)as X→∞.□ Acknowledgements We would like to thank Professor Zeev Rudnick for suggesting that the results of Nair and Tenenbaum could be used to prove the main result of this paper. We would like to thank an anonymous referee whose comments and suggestions helped to improve the presentation of the paper. The first author is grateful to the Leverhulme Trust (RPG-2017-320) for the support through the research grant “Moments of L-functions in Function Fields and Random Matrix Theory”. References 1 H. Delange, Sur les formules dues á Atle Selberg, Bull. Sci. Math. 2 série 83 ( 1959), 101– 111. 2 P. Erdős and R. R. Hall, Values of the divisor function on short intervals, J. Number Theory 12 ( 1980), 176– 187. Google Scholar CrossRef Search ADS 3 I. Kátai, On the local behaviour of the function d(n) (Hungarian.), Mat. Lapok 18 ( 1967), 297– 302. 4 M. Nair and G. Tenenbaum, Short sums of certain arithmetic functions, Acta Math. 180 ( 1998), 119– 144. Google Scholar CrossRef Search ADS 5 A. Selberg, Note on the paper by L. G. Sathe, J. Indian Math. Soc. 18 ( 1954), 83– 87. 6 E. C. Titchmarsh, The Theory of the Riemann Zeta-function , 2nd edn, Oxford University Press, New York, 1986. © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

The Quarterly Journal of Mathematics – Oxford University Press

**Published: ** Mar 2, 2018

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