An explicit P\'{o}lya-Vinogradov inequality via Partial Gaussian sums

TL;DR

This paper derives a new explicit constant for the Pólya-Vinogradov inequality for squarefree moduli, improving bounds by using partial Gaussian sums instead of traditional exponential sum methods.

Contribution

It introduces a novel approach using partial Gaussian sums to obtain explicit bounds for the Pólya-Vinogradov inequality, surpassing previous methods for large moduli.

Findings

Explicit constants for even and odd characters derived

Improved bounds for large squarefree moduli

Power saving on minor arcs achieved

Abstract

In this paper we obtain a new fully explicit constant for the P\'olya-Vinogradov inequality for squarefree modulus. Given a primitive character $\chi$ to squarefree modulus $q$, we prove the following upper bound \begin{align*} \left| \sum_{1 \le n\le N} \chi(n) \right|\le c \sqrt{q} \log q, \end{align*} where $c=1/(2\pi^2)+o(1)$ for even characters and $c=1/(4\pi)+o(1)$ for odd characters, with an explicit $o(1)$ term. This improves a result of Frolenkov and Soundararajan for large $q$. We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of $\log{q}$ as in previous approaches and is an important factor for fully explicit bounds.