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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.02612 |
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| _version_ | 1866909940252147712 |
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| author | Mistiaen, Ludovic |
| author_facet | Mistiaen, Ludovic |
| contents | In this paper, for a given Dirichlet character mod $N$ with $4\nmid N$, we give a lower bound of order $\sqrt{s/\log(s)}$ for the dimension of the $\mathbb{Q}(e^{2iπ/N})$-vector space spanned by the values of its $L$-function at integers $\leq s$ of a given parity. We thus generalize a result Fischler proved in 2021, corresponding to the principal character mod 1. To this end, we construct linear combinations of these values of $L$-function with a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on a ``Shidlovskii's lemma''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear independence of values of Dirichlet $L$ functions Mistiaen, Ludovic Number Theory In this paper, for a given Dirichlet character mod $N$ with $4\nmid N$, we give a lower bound of order $\sqrt{s/\log(s)}$ for the dimension of the $\mathbb{Q}(e^{2iπ/N})$-vector space spanned by the values of its $L$-function at integers $\leq s$ of a given parity. We thus generalize a result Fischler proved in 2021, corresponding to the principal character mod 1. To this end, we construct linear combinations of these values of $L$-function with a refined version of Siegel's lemma, and we apply to them a linear independence criterion generalizing the one used by Fischler. To check the assumptions of this criterion, we rely on a ``Shidlovskii's lemma''. |
| title | Linear independence of values of Dirichlet $L$ functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.02612 |