On the mean square average of Dirichlet $L$-function over characters of odd parityin a special case

Abstract

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