Zeros of $L$-functions and large partial sums of Dirichlet coefficients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917743474769920 |
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| author | Kerr, Bryce Klurman, Oleksiy Thorner, Jesse |
| author_facet | Kerr, Bryce Klurman, Oleksiy Thorner, Jesse |
| contents | Let $L(s,π)=\sum_{n=1}^{\infty}λ_π(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $λ_π(n)$ strongly repel the low-lying zeros of $L(s,π)$ away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zeros of $L$-functions and large partial sums of Dirichlet coefficients Kerr, Bryce Klurman, Oleksiy Thorner, Jesse Number Theory 11L40, 11N56, 11F66 Let $L(s,π)=\sum_{n=1}^{\infty}λ_π(n)n^{-s}$ be an $L$-function that satisfies a weak form of the generalized Ramanujan conjecture. We prove that large partial sums of $λ_π(n)$ strongly repel the low-lying zeros of $L(s,π)$ away from the critical line. Our results extend and quantitatively improve preceding work of Granville and Soundararajan. |
| title | Zeros of $L$-functions and large partial sums of Dirichlet coefficients |
| topic | Number Theory 11L40, 11N56, 11F66 |
| url | https://arxiv.org/abs/2408.03938 |