Multiplicative mimicry and improvements to the Pólya-Vinogradov inequality
type: Item
instance of: scholarly article
title: Multiplicative mimicry and improvements to the Pólya-Vinogradov inequality
zbMATH Open document ID: 1263.11076
author: Leo Goldmakher
published in: Algebra \& Number Theory
publication date: 2012-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.5547
full work available at URL: p04.xhtml
reviewed by: Angel V. Kumchev
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11L03
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 6064706
zbMATH Keywords: Dirichlet characters
zbMATH Keywords: character sums
zbMATH Keywords: exponential sums
zbMATH Keywords: multiplicative functions
MaRDI profile type: Publication
OpenAlex ID: W2031595800
arXiv ID: 0911.5547
DOI: 10.2140/ANT.2012.6.123
review text: Let \(\chi\) be a primitive Dirichlet character modulo \(q\), and write NEWLINE\[NEWLINE M(\chi) = q^{-1/2} \max_{x \leq q} \Big| \sum_{n \leq x} \chi(n) \Big|. NEWLINE\]NEWLINE The classical Pólya--Vinogradov inequality states that \(M(\chi) \ll \log q\). Improvements on this result have long attracted the attention of analytic number theorists, yet have been notoriously difficult to obtain. In 1977, \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Invent. Math. 43, 69--82 (1977; Zbl 0362.10036)] proved, conditionally on the Generalized Riemann Hypothesis (GRH), that \(M(\chi) \ll \log\log q\). It has been known since the 1930's that this bound is best-possible for quadratic characters. In 2007, \textit{A. Granville} and \textit{K. Soundararajan} [J. Am. Math. Soc. 20, 357--384 (2007; Zbl 1210.11090)] proved that the Montgomery-Vaughan bound is, in fact, best-possible for characters of any even order. Further, for characters of odd order, Granville and Soundararajan obtained the first unconditional improvement on the Pólya-Vinogradov inequality: If \(\chi\) has an odd order \(g > 1\), then NEWLINE\[NEWLINE M(\chi) \ll (\log q)^{1-\delta_g/2 + o(1)}, NEWLINE\]NEWLINE where \(\delta_g = 1 - (g/\pi)\sin(\pi/g)\). Under the assumption of GRH, they also obtained a similar improvement on the Montgomery--Vaughan bound for characters of odd order.NEWLINENEWLINEIn the paper under review, the author develops further the technique of Granville and Soundararajan and proves that if \(\chi\) has an odd order \(g > 1\), then NEWLINE\[NEWLINE M(\chi) \ll (\log q)^{1-\delta_g + o(1)}, NEWLINE\]NEWLINE with \(\delta_g\) as above. Further, under the assumption of GRH, he shows that NEWLINE\[NEWLINE (\log\log q)^{1 - \delta_g - o(1)} \ll M(\chi) \ll (\log\log q)^{1-\delta_g + o(1)}. NEWLINE\]NEWLINE The author derives these results from a delicate (and rather technical) bound for the exponential sum NEWLINE\[NEWLINE \sum_{n \leq x} \frac {f(n)}{n} e(n\alpha), NEWLINE\]NEWLINE whe…
description: scientific article; zbMATH DE number 6064706