On two mean square averages of Dirichlet $L$-function

TL;DR

This paper explicitly evaluates the mean square averages of Dirichlet L-functions at specific points, using trigonometric sums and Bernoulli polynomials, and expresses results in terms of classical arithmetic functions.

Contribution

It provides explicit formulas for the mean square averages of L(3,χ) and L(4,χ) over characters of same parity, advancing understanding in analytic number theory.

Findings

Explicit formulas for mean squares of L(3,χ) and L(4,χ)

Use of trigonometric sums and Bernoulli polynomials in evaluation

Expressions involve Euler and Jordan totient functions

Abstract

Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,\chi)$ and $L(4,\chi)$ using certain trigonometric sums and Bernoulli polynomials and express them in terms of the Euler totient function $\phi$ and the Jordan totient function $J_s$.