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.
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We study properties of the signature function of the torus knot Tp,g. First we provide a very elementary proof of the formula for the integral of the signature over the circle. We also obtain a closed formula for the Tristram-Levine signature of a torus knot in terms of Dedekind sums.
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