The divisor function on residue classes II

ACTA ARITHMETICA Online First version

The divisor function on residue classes II by

Prapanpong Pongsriiam (Nakhon Pathom) and Robert C. Vaughan (University Park, PA) 1. Introduction and main results. Let ψ(x, q, a) denote the usual function of prime number theory. There is a long history of estimates for 2 q X X x ψ(x, q, a) − (1.1) ϕ(q) q≤Q

a=1 (a,q)=1

going back to at least Barban [Bar] (at least in the case Q = x), Davenport and Halberstam [DH], and Gallagher [Ga], and culminating in the asymptotic formula of Montgomery [Mon1] and its refinement by Hooley [Ho1, Ho2]. Further developments occur in Friedlander and Goldston [FG] and Goldston and Vaughan [GV], and the methods have been broadened by Hooley [Ho3], [Ho4]–[Ho8] and Vaughan [Va1, Va2] to study a wide range of sequences. The general principle is that the underlying arithmetical function, for example in the case of primes the von Mangoldt function, should satisfy an analogue of the Siegel–Walfisz theorem and there should be some understanding of the behavior of its `2 mean. In particular it should be constant on average, although not necessarily in `2 mean, but the nature of the asymptotic formula obtained depends on the `2 mean. The function r(n), the number of representations of n as a sum of two squares, does not fall within the orbit of the general results, since whilst the average of r(n) is a constant, that of r(n)2 is large. Yet Dancs [Dan] has shown that there is an asymptotic formula for q 2 XX X (1.2) r(n) − xf (q, a) q≤Q a=1

n≤x n≡a (mod q)

where f (q, a) is an appropriate arithmetical function. 2010 Mathematics Subject Classification: Primary 11N37; Secondary 11A25, 11B25. Key words and phrases: divisor function, residue class, arithmetic progression. Received 13 December 2016; revised 25 August 2017. Published online *. DOI: 10.4064/aa161213-24-10

[1]

c Instytut Matematyczny PAN, 2018

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P. Pongsriiam and R. C. Vaughan

The purpose here is to investigate the corresponding expression in the case of d(n), the number of positive divisors of n, that is, X (1.3) A(x, q, a) = d(n). n≤x n≡a (mod q)

This has some general interest since it is as an example in which the underlying function is not bounded on average. The function A(x, q, a) has been widely studied: see for example [FoI, FrI1, FrI2, Ch, Sm, LE, Ma, Nak, No, He]. However, as far as we are aware, only Motohashi [Mot] has studied q XX (1.4) V (x, Q) = |A(x, q, a) − M (x, q, a)|2 q≤Q a=1

for an appropriate approximating function M (x, q, a) and then only in the case Q = x. He gives [Mot, p. 177] (1.5)

V (x, x) =

x2 (log x)3 + g2 x2 (log x)2 + g1 x2 log x + g0 x2 π2 + O(x15/8 (log x)2 ),

where g0 , g1 , g2 are absolute constants. The function M (x, q, a) represents a considerable challenge to the investigation since not only does it depend in a non-trivial way on a as well as q but it does not split simply as F (x)G(q, a). For our purposes the most convenient general form for M (x, q, a) is   x X ct (a) x (1.6) M (x, q, a) = log 2 + 2γ − 1 q t t t|q

as introduced in [PV], which leads to some simplifications in what follows. Here ct (a) is Ramanujan’s sum   t X ba . (1.7) ct (a) = e t b=1 (b,t)=1

In this article, we extend (1.5) to the general case Q ≤ x and with a substantially smaller error term. This is also a significant improvement on the results in the first author’s thesis [P1]. Theorem 1.1. Let ε > 0. Then x2 (1.8) V (x, x) = 2 (log x)3 + g2 x2 (log x)2 + g1 x2 log x + g0 x2 + O(x5/3+ε ), π and for 1 ≤ Q < x and 0 < Θ < 1, we have

The divisor function on residue classes II

(1.9)

3

    Qx Q2 3 (6γ − 3 − 6 log 2π) Q2 2 + Qx log V (x, Q) = 2 log π x π2 x 2 Q + g3 Qx log + g4 Qx log x + g5 Qx x + O(x5/3+ε + Q1+Θ x1−Θ (log x)2 ),

where γ is the Euler constant, γ1 is the Stieltjes constant given by  X log m (log n)2 − , γ1 = − lim n→∞ m 2 m≤n

and
3 4γ − 4γ 2 − π 2 /3 − 6 + (8γ − 4) log 2π + 8ζ 00 (0) , 2 π and the constants g0 , g1 , g2 , g4 , g5 are absolute and can be computed explicitly. g3 = −

We remark that we do not give the values of the constants g0 , g1 , g2 , g4 , g5 explicitly since they do not lead to any obvious simplifications in the main terms. As is usual with results of this kind, there is a loss of uniformity as Q approaches x due to the fact that for q close to x there are very few, maybe no, terms in A(x, q, a). Thus we cannot simply obtain V (x, x) from V (x, Q) by substituting Q = x. Generally, as is observed in the case of Λ(n) in Goldston and Vaughan [GV], for each fixed k there will be a different asymptotic formula valid for x/(k + 1) < Q ≤ x/k. Blomer [Bl] has established (in a different notation) that (1.10)

q X

(A(x, q, a) − M (x, q, a))2  x1+ε

a=1

and Banks, Heath-Brown, and Shparlinski [BHS, Theorem 3.1] give bounds for q X (1.11) |A(x, q, a) − M (x, q, a)|. a=1 (a,q)=1

However, it is readily seen from an application of the Cauchy–Schwarz inequality that Blomer’s bound is always stronger. Note also that for (a, q) = 1, the expected value of A(x, q, a) in [BHS] is given in a different form but it actually is the same, after correcting a sign error in [BHS, (1.4), (2.24)]. Let     y 2 3 6γ − 3 − 6 log 2π y2 2 yx (1.12) + yx log F (x, y) = 2 log π x π2 x 2 y + g3 yx log + g4 yx log x + g5 yx, x

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P. Pongsriiam and R. C. Vaughan

(1.13)

G(x, y) =

∂F (x, y). ∂y

Then it seems quite plausible that q X

(1.14)

(A(x, q, a) − M (x, q, a))2 ∼ G(x, q)

a=1

with a reasonable error term for some range of q and x. This might be beyond current technology but probably it can be established for almost all q in some range. To prove Theorem 1.1, we follow the method developed by Hooley and divide the proof into four steps. We define, for R < Q ≤ x, the following quantities: (1.15)

V (x, R, Q) =

(1.16)

S1 = S1 (x, R, Q) =

(1.17)

S2 = S2 (x, R, Q) =

(1.18)

S3 = S3 (x, R, Q) =

q X X

(A(x, q, a) − M (x, q, a))2 ,

R