On the mean values of Dirichlet L-functions

Abstract

Let p>3 be a prime, χ be a Dirichlet character modulo p, and let L(s,χ) denote the Dirichlet L-function. Let n≥1 be an integer. In this paper we study the mean values of2p−1∑χmodpχ(−1)=−1χ(c)L(1,χ)L(n,χ¯)and2p−1∑χmodpχ(−1)=1χ(c)L(2,χ)L(n,χ¯) for c=1,2,3,4, and gave a few identities, by using short interval character sums and Bernoulli polynomials.