On the constant in the Pólya-Vinogradov inequality

Abstract

In this paper we obtain a new constant in the Pólya-Vinogradov inequality. Our argument follows previously established techniques which use the Fourier expansion of an interval to reduce to Gauss sums. Our improvement comes from approximating an interval by a function with slower decay on the edges and this allows for a better estimate of the ${\ell }_1$ norm of the Fourier transform. This approximation induces an error for our original sums which we deal with by combining some ideas of Hildebrand with Garaev and Karatsuba concerning long character sums.

Introduction

Given integers $q,M$ and N and a primitive multiplicative character χ mod q we consider estimating the sums

$$S(\chi ,M,N)=\sum _{M<n⩽M+N}\chi (n),$$

and when $M=0$ we write

$$S(\chi ,0,N)=S(\chi ,N).$$

The first nontrivial result in this direction is due to Pólya and Vinogradov from the early 1900's and states that

$$S(\chi ,M,N)⩽c{q}^{1/2}\log q,$$

for some constant c independent of q. Up to improvements in the constant c this bound has remained sharpest known for the past 100 years and a fundamental question in the area of character sums is whether c can be taken arbitrarily small. Montgomery and Vaughan [19] have shown conditionally on the Generalized Riemann Hypothesis that

$$S(\chi ,M,N)\ll q^{1/2}\log \log q.$$

This would be best possible since Payley [20] has shown that there exists an infinite sequence of integers q and characters χ mod q such that

$$\underset{1⩽N<q}{\max }S(\chi ,N)\gg q^{1/2}\log \log q.$$

Although making a $o(1)$ improvement on the Pólya-Vinogradov inequality for all characters χ and intervals $(M,M+N]$ remains an open problem, there has been progress in determining general situations where such improvements can be made. Concerning short character sums, a classic result of Burgess [5], [6] states that for any primitive χ

$$|S(\chi ,M,N)|⩽N^{1-1/r}{q}^{(r+1)/(4r^2)+o(1)},$$

provided $r⩽3$ and for any $r⩾2$ if q is cubefree. Hildebrand [15] has shown that one can improve on the constant in the Pólya-Vinogradov inequality given estimates for short character sums and Bober and Goldmakher [2] and Fromm and Goldmakher [10] have shown how improvements on the constant in the Polya-Vinogradov inequality may be used to obtain new estimates for short character sums.

Concerning long character sums, Hildebrand [16] has shown that if $\chi (-1)=1$ then

$$|S(\chi ,\alpha q)|<\epsilon q^{1/2}\log q,$$

for all $\alpha \in (0,1)$ except for a set of measure $q^{-c_1\epsilon }$ and that if $\alpha =o(1)$ and $\chi (-1)=1$ then

$$S(\chi ,\alpha q)=o(q^{1/2}\log q).$$

Bober and Goldmakher [1] and Bober, Goldmakher, Granville and Koukoulopoulos [3] have obtained much more precise results concerning the distribution of long character sums and Granville and Soundararajan [13] have obtained results concerning the distribution of short character sums. Granville and Soundararajan [14] have also shown that

$$S(\chi ,N)\ll q^{1/2}{(\log q)}^{1-{\delta }_g/2+o(1)},$$

if χ has odd order g, where

$${\delta }_g=1-\frac{g}{\pi }\sin \frac{\pi }{g},$$

and the factor ${\delta }_g/2$ occurring above has been improved by Goldmakher [12] and Lamzouri and Mangerel [17].

We consider the problem of estimating $S(\chi ,M,N)$ uniformly over $\chi ,M$ and N in the Pólya-Vinogradov range. Since the work of Pólya and Vinogradov there have been a number of improvements to the constant c occurring in (1). The sharpest constant is due to Pomerance [21] and is based on ideas of Landau [18] and an unpublished observation of Bateman, see [15]. In particular, Pomerance [21, Theorem 1] shows that

$$|S(\chi ,M,N)|⩽\{\begin{array}{c}(\frac{2}{{\pi }^2}+o(1))q^{1/2}\log q,\text{if}\chi (-1)=1,\hfill \\ (\frac{1}{2\pi }+o(1))q^{1/2}\log q,\text{if}\chi (-1)=-1.\hfill \end{array}$$

Pomerance gives the lower order terms explicitly and these have been improved by Frolenkov [8] and Frolenkov and Soundararajan [9]. In the case of intervals starting from the origin one may obtain better constants with the sharpest given by Granville and Soundararajan [14], improving on previous results of Hildebrand [15], [16]. We note that both [14] and [15] obtain sharper constants than our main result although these are restricted to sums of the form $S(\chi ,0,N)$. The strength of our result lies in the estimation of $S(\chi ,M,N)$ for arbitrary M.

In this paper we obtain a new constant in the Pólya-Vinogradov inequality for arbitrary intervals. Our argument follows previously established techniques which use the Fourier expansion of an interval to reduce to Gauss sums. Our improvement comes from approximating an interval by a function with slower decay on the edges which allows for a better estimate of the ${\ell }_1$ norm of the Fourier transform. This induces an error for our original sums which we deal with by combining some ideas of Hildebrand [15] with Garaev and Karatsuba [11]. A new feature of our argument is that we use estimates for long character sums to improve on the constant in the Pólya-Vinogradov inequality. For example, if one could show that for any $\epsilon >0$ we have

$$S(\chi ,M,N)=o(q^{1/2}\log q),$$

for arbitrary M whenever $N<q^{1-\epsilon }$ and sufficiently large q then it would follow from our argument that

$$S(\chi ,M,N)=o(q^{1/2}\log q),$$

for arbitrary M and N.

We end the introduction by mentioning some notation and conventions used throughout. Given a real number x we let

$$e(x)=e^{2\pi ix},$$

and for integer q define

$$e_q(x)=e(x/q).$$

We will adopt the following convention to avoid ambiguity in order of operations when fractional indices occur as exponents. Given a real number q and integers ${\alpha }_1,\dots ,{\alpha }_j,{\beta }_1,\dots ,{\beta }_j$ we write

$$q^{{\alpha }_1\dots {\alpha }_k/{\beta }_1\dots {\beta }_k}=q^{\frac{{\alpha }_1\dots {\alpha }_k}{{\beta }_1\dots {\beta }_k}}.$$

In general, if a term of the form

$${\alpha }_1\dots {\alpha }_k/{\beta }_1\dots {\beta }_k,$$

occurs in the exponent of some real number, we take the order of operations as all multiplications occur before the / symbol and all divisions occur after the / symbol.