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

Louboutin S.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 2/3

165--174

Explicit formulas for the quadratic mean value at s = 1 of the Dirichlet L-functions associated with the set Xf¯ of the ϕ(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 Xf¯ 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.