Quartic and Octic Characters Modulo n

TL;DR

This paper investigates the asymptotic behavior of primitive quartic and octic characters modulo n, revealing growth patterns with logarithmic factors, contrasting with quadratic and cubic cases.

Contribution

It provides the first detailed asymptotic analysis of primitive quartic and octic characters, including leading coefficients, showing their growth rates as n increases.

Findings

Primitive quartic characters grow with ln(n)

Primitive octic characters grow with (ln(n))^2

Leading coefficients in asymptotic formulas are computed

Abstract

The average number of primitive quadratic Dirichlet characters of modulus n tends to a constant as n->infty. The same is true for primitive cubic characters. It is therefore surprising that, as n->infty, the average number of primitive quartic characters of modulus n grows with ln(n), and that the average number of primitive octic characters of modulus n grows with ln(n)^2. Leading coefficients in the asymptotic expressions are also computed.