On the mean square average of special values of L-functions

Abstract

Let chi be a Dirichlet character and L(s, chi) be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordan's and Euler's totient function for the mean square average of L(1, chi) when chi ranges over all odd characters modulo k and L(2, chi) when x ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of L(r, chi) if chi and r >= 1 have the same parity and chi ranges over all odd (or even) characters modulo k, though the required calculations become formidable when r >= 3. Consequently, we see that for almost all odd characters modulo k, vertical bar L(1, chi)vertical bar < phi(k), where phi(x) is any function monotonically tending to infinity. (C) 2011 Elsevier Inc. All rights reserved.