The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

TL;DR

This paper studies the distribution of maximum partial sums of certain periodic complex functions, including Kloosterman sums, providing precise estimates and constructing examples where classical bounds are sharp.

Contribution

It offers uniform estimates for the distribution of maximum partial sums of trace functions, extending previous results and constructing families with sharp Pólya-Vinogradov bounds.

Findings

Uniform estimates for distribution functions of maximum partial sums

Application to Kloosterman sums and trace functions

Construction of families with sharp Pólya-Vinogradov bounds

Abstract

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$-periodic complex valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$-adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$-periodic complex valued functions which satisfy our conditions, but for which the P\'olya-Vinogradov inequality is sharp.