On the second smallest prime non-residue

TL;DR

This paper provides improved explicit upper bounds on the second smallest prime non-residue of a Dirichlet character modulo a prime, with applications to norm-Euclidean Galois fields.

Contribution

It introduces new explicit bounds on the second smallest prime non-residue and estimates on their product, advancing understanding in number theory.

Findings

Improved upper bounds on $q_2$ for prime non-residues.

Explicit estimates on the product $q_1 q_2$.

Applications to norm-Euclidean Galois fields.

Abstract

Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. Let $q_1<q_2$ denote the two smallest prime non-residues of $\chi$. We give explicit upper bounds on $q_2$ that improve upon all known results. We also provide a good upper estimate on the product $q_1 q_2$ which has an upcoming application to the study of norm-Euclidean Galois fields.