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Article Dans Une Revue Canadian Journal of Mathematics = Journal Canadien de Mathématiques Année : 2023

Mean square values of L-functions over subgroups for non primitive characters, Dedekind sums and bounds on relative class numbers

Résumé

An explicit formula for the mean value of |L(1, χ)| 2 is known, where χ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field Q(ζ p) follow. Lately the authors obtained that the mean value of |L(1, χ)| 2 is asymptotic to π 2 /6, where χ runs over all odd primitive Dirichlet characters of prime conductors p ≡ 1 (mod 2d) which are trivial on a subgroup H of odd order d of the multiplicative group (Z/pZ) * , provided that d log p log log p. Bounds on the relative class number of the subfield of degree p−1 2d of the cyclotomic field Q(ζ p) follow. Here, for a given integer d 0 > 1 we consider the same questions for the nonprimitive odd Dirichlet characters χ modulo d 0 p induced by the odd primitive characters χ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of |L(1, χ)| 2 is asymptotic to π 2 6 q|d0 1 − 1 q 2 , where χ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order d log p log log p. As a consequence we improve the previous bounds on the relative class number of the subfield of degree p−1 2d of the cyclotomic field Q(ζ p). Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.
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Dates et versions

hal-03983361 , version 1 (10-02-2023)

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Stéphane Louboutin, Marc Munsch. Mean square values of L-functions over subgroups for non primitive characters, Dedekind sums and bounds on relative class numbers. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, In press, 75 (5), pp.1711-1743. ⟨10.4153/S0008414X2300010X⟩. ⟨hal-03983361⟩
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