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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.02802 |
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| _version_ | 1866929374554488832 |
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| author | Louboutin, Stéphane |
| author_facet | Louboutin, Stéphane |
| contents | An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo not necessarily prime moduli $f>2$ that are trivial on a subgroup $H$ of the multiplicative group $({\mathbb Z}/f{\mathbb Z})^*$. This explicit formula involves summation $S(H,f)$ of Dedekind sums $s(h,f)$ over the $h\in H$. A result on some cancelation of the denominators of the $s(h,f)$'s when computing $S(H,f)$ is known. Here, we prove that for some explicit families of $f$'s and $H$'s this known result on cancelation of denominators is the best result one can expect. Finally, we surprisingly prove that for $p$ a prime, $m\geq 2$ and $1\leq n\leq m/2$, the values of the Dedekind sums $s(h,p^m)$ do not depend on $h$ as $h$ runs over the elements of order $p^n$ of the multiplicative cyclic group $({\mathbb Z}/p^m{\mathbb Z})^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dedekind sums and mean square value of $L(1,χ)$ over subgroups Louboutin, Stéphane Number Theory An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo not necessarily prime moduli $f>2$ that are trivial on a subgroup $H$ of the multiplicative group $({\mathbb Z}/f{\mathbb Z})^*$. This explicit formula involves summation $S(H,f)$ of Dedekind sums $s(h,f)$ over the $h\in H$. A result on some cancelation of the denominators of the $s(h,f)$'s when computing $S(H,f)$ is known. Here, we prove that for some explicit families of $f$'s and $H$'s this known result on cancelation of denominators is the best result one can expect. Finally, we surprisingly prove that for $p$ a prime, $m\geq 2$ and $1\leq n\leq m/2$, the values of the Dedekind sums $s(h,p^m)$ do not depend on $h$ as $h$ runs over the elements of order $p^n$ of the multiplicative cyclic group $({\mathbb Z}/p^m{\mathbb Z})^*$. |
| title | Dedekind sums and mean square value of $L(1,χ)$ over subgroups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2406.02802 |