Dedekind sums and mean square value of L(1,χ) over subgroups

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 ( Z/f Z)^*. This explicit formula involves summation $S(H,f)$ of Dedekind sums $s(h,f)$ over the h∈ 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≥ 2 and 1≤ n≤ m/2, the values of the Dedekind sums s(h,pᵐ) do not depend on h as h runs over the elements of order pⁿ of the multiplicative cyclic group ( Z/pᵐ Z)^*.