On the constant in the Polya-Vinogradov inequality

TL;DR

This paper improves the constant in the Pólya-Vinogradov inequality by refining the Fourier analysis approach and better estimating character sums, leading to tighter bounds in number theory.

Contribution

It introduces a novel approximation method for intervals that enhances the Fourier transform estimate, resulting in a better constant in the inequality.

Findings

New constant in the Pólya-Vinogradov inequality established

Enhanced Fourier analysis technique developed for character sums

Improved bounds on long character sums achieved

Abstract

In this paper we obtain a new constant in the P\'{o}lya-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.