A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910959348482048 |
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| author | Banks, William D. |
| author_facet | Banks, William D. |
| contents | For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. We show that $RH_{sim}^\dagger[X]$ (for any $X$) is a consequence of the generalized Riemann hypothesis.
Assuming only the generalized Lindelöf hypothesis, we show that if $RH_{sim}^\dagger[X]$ holds for one primitive character $X$, then it holds for every such $X$. If this occurs, then for every character $χ$ (primitive or not), all simple zeros of $L(s,χ)$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11605 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions Banks, William D. Number Theory Primary: 11M06, 11M26, Secondary: 11M20 For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. We show that $RH_{sim}^\dagger[X]$ (for any $X$) is a consequence of the generalized Riemann hypothesis. Assuming only the generalized Lindelöf hypothesis, we show that if $RH_{sim}^\dagger[X]$ holds for one primitive character $X$, then it holds for every such $X$. If this occurs, then for every character $χ$ (primitive or not), all simple zeros of $L(s,χ)$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation. |
| title | A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions |
| topic | Number Theory Primary: 11M06, 11M26, Secondary: 11M20 |
| url | https://arxiv.org/abs/2410.11605 |