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Article Dans Une Revue Functiones et Approximatio Commentarii Mathematici Année : 2023

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Résumé

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.
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Dates et versions

hal-04057621 , version 1 (04-04-2023)

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Stéphane R Louboutin. Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers. Functiones et Approximatio Commentarii Mathematici, 2023, 68 (1), ⟨10.7169/facm/2027⟩. ⟨hal-04057621⟩
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