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

Abstract: Let χ be a Dirichlet character and be its L-function. Using weighted averages of Gauss and Ramanujan sums, we find exact formulas involving Jordanʼs and Eulerʼs totient function for the mean square average of when χ ranges over all odd characters modulo k and when χ ranges over all even characters modulo k. In principle, using our method, it is always possible to find the mean square average of if χ and have the same parity and χ ranges over all odd (or even) characters modulo k, though the required calculations become formidable when . Consequently, we see that for almost all odd characters modulo k, , where is any function monotonically tending to infinity. [Copyright &y& Elsevier]