The average least character nonresidue and further variations on a theme of Erdos

TL;DR

This paper investigates the average least nonresidue for Dirichlet characters and the least prime not splitting completely in cubic fields, revealing specific limiting constants and their behavior over various families.

Contribution

It establishes new average results for least nonresidues and splitting primes, connecting these to explicit constants and extending previous understanding in number theory.

Findings

Average least nonresidue approaches the least prime not dividing q.

Average over cubic fields converges to a specific constant.

Provides explicit limiting values for these averages.

Abstract

For each nonprincipal Dirichlet character $\chi$, let $n_\chi$ be the least $n$ with $\chi(n) \notin \{0,1\}$. We show that as the average of $n_\chi$ over all nonprincipal characters $\chi$ modulo $q$ is $\ell(q) + o(1)$, where $\ell(q)$ denotes the least prime not dividing $q$. Moreover, if one averages over all nonprincipal characters of modulus at most $x$, the average approaches a particular limiting value 2.5350541804. We also prove a result of this type for cubic number fields: If one averages over all cubic fields $K$, ordered by the absolute value of their discriminant, then the mean value of the least rational prime that does not split completely in $K$ is another particular constant 2.1211027269.