Dedekind Sums, Mean Square Value of $L$-Functions at $s=1$ and Upper Bounds on Relative Class Numbers

Explicit formulas for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the set $X_f^-$ of the $\phi (f)/2$ odd Dirichlet characters mod $f$ are known. They have been used to obtain explicit upper bounds for relative class numbers of cyclotomic number fields. Here we present a generalization of these results: we show that explicit formulas for quadratic mean values at $s=1$ of Dirichlet $L$-functions associated with subsets of $X_f^-$ can be obtained. As an application we use them to obtain explicit upper bounds for relative class numbers of imaginary subfields of cyclotomic number fields.