March 2023 Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers
Stéphane R. Louboutin
Funct. Approx. Comment. Math. 68(1): 101-112 (March 2023). DOI: 10.7169/facm/2027

Abstract

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.

Citation

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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." Funct. Approx. Comment. Math. 68 (1) 101 - 112, March 2023. https://doi.org/10.7169/facm/2027

Information

Published: March 2023
First available in Project Euclid: 15 December 2022

MathSciNet: MR4564865
Digital Object Identifier: 10.7169/facm/2027

Subjects:
Primary: 11F20 , 11M20 , 11R18 , 11R29

Keywords: $L$-function , cyclotomic field , Dedekind sums , Dirichlet character , mean square value , relative class number

Rights: Copyright © 2023 Adam Mickiewicz University

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Vol.68 • No. 1 • March 2023
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