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

TL;DR

This paper derives explicit formulas for the mean square value of certain Dirichlet L-functions at s=1, involving Dedekind sums, and explores their properties and cancellations for specific moduli and subgroups.

Contribution

It extends known explicit formulas for mean values of Dirichlet L-functions to non-prime moduli and analyzes Dedekind sums over subgroups, revealing new invariance properties.

Findings

Explicit formulas involve Dedekind sums over subgroups.

Known denominator cancellation results are optimal for certain families.

Dedekind sums are constant over elements of fixed order in cyclic groups.

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 $({\mathbb Z}/p^m{\mathbb Z})^*$.