Conditional upper bounds on the least character non-residue

TL;DR

This paper develops new conditional upper bounds for the smallest character non-residue by extending methods related to zero-free regions of L-functions, connecting to prior work on character sums.

Contribution

It introduces extended techniques to establish bounds on least character non-residues based on zero-free regions within the critical strip.

Findings

Derived upper bounds contingent on zero-free regions

Connected bounds to recent results on character sums

Extended methods applicable to arbitrary heights in the critical strip

Abstract

We extend known methods to establish upper bounds on the least character non-residues contingent on different zero-free regions within the critical strip, in particular on bounded rectangles within the critical strip along the $\sigma=1$ line at arbitrary heights. This relates to earlier conditional results on least character non-residues, and recent results of Granville and Soundararajan on character sums.