Averages of values of 𝐿-series

Abstract

We obtain an exact formula for the average of values of L L -series over two independent odd characters. The average of any positive moment of values at s = 1 s=1 is then expressed in terms of finite cotangent sums subject to congruence conditions. As consequences, bounds on such cotangent sums, limit points for the average of first moment of L L -series at s = 1 s=1 and the average size of positive moments of character sums related to the class number are deduced.