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

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 $\varphi$ and the Jordan totient function $J_s$.