On the mean values of products of Dirichlet $L$-functions at positive integers

In this paper, we study the mean value distributions of Dirichlet $L$-functions at positive integers. We give some explicit formulas for the mean values of products of two and three Dirichlet $L$-functions at positive integers weighted by Dirichlet characters that involve the Bernoulli functions and Jordan’s totient functions. The results presented here are the generalizations of various known formulas.