Dedekind sums and mean square value of $L(1,\chi)$ over subgroups

Abstract

An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo not necessarily prime moduli $f>2$ that are trivial on a subgroup $H$ of the multiplicative group $({\mathbb Z}/f{\mathbb Z})^*$. This explicit formula involves summation $S(H,f)$ of Dedekind sums $s(h,f)$ over the $h\in H$. A result on some cancelation of the denominators of the $s(h,f)$'s when computing $S(H,f)$ is known. Here, we prove that for some explicit families of $f$'s and $H$'s this known result on cancelation of denominators is the best result one can expect. Finally, we surprisingly prove that for $p$ a prime, $m\geq 2$ and $1\leq n\leq m/2$, the values of the Dedekind sums $s(h,p^m)$ do not depend on $h$ as $h$ runs over the elements of order $p^n$ of the multiplicative cyclic group $({\mathbb Z}/p^m{\mathbb Z})^*$.