The discrete mean square of Dirichlet $L$-function at integral arguments

Abstract

In this paper we shall make complete structural elucidation of the explicit formula for the (discrete) mean square of Dirichlet $L$-function at integral arguments, save for the case $s=1$, this being completely settled in [1] recently. We shall treat the cases of negative and positive integers arguments separately, the former case being a preliminary and inclusive in the second. It will turn out that in respective cases the characteristic difference properties of Bernoulli polynomials and of the Hurwitz zeta-function are essential and telescoping the resulting difference equations, we obtain the results, revealing the underlying simple structure (known before 1905 at least).