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2 Bulletin of the Polish Academy of Sciences. Mathematics

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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.

The analytic rank of a family of jacobians of Fermat curves

Jędrzejak T.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 3-4

199-206

We study the family of curves Fm(p) : xp + yp = m, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves Fm(p). As a corollary we conclude that the Jacobians of the curves Fm(5) with even analytic rank and those with odd analytic rank are equally distributed.