Dedekind sums and mean square value of L (1, ) 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) ^*.