Dedekind sums, mean square value of L-functions at s = 1 and upper bounds on relative class numbers. A survey and open problems

International audienceExplicit formulas for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the set X − f of the φ(f)/2 odd Dirichlet characters modulo 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 obtain explicit upper bounds for relative class numbers of imaginary subfields of cyclotomic number fields