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

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.

19 pages

References in corpus (4)

• Secondary terms in counting functions for cubic fields
• Counting Cubic Extensions with given Quadratic Resolvent
• On the Davenport-Heilbronn theorems and second order terms
• The smallest prime that does not split completely in a number field

Cited by in corpus (3)

• The shape of cubic fields
• Mean-Value of Product of Shifted Multiplicative Functions and Average Number of Points on Elliptic Curves
• Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$