Conditional upper bounds on the least character non-residue

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 $\ddot{I}ƒ=1$ 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.