A metric approach to zero-free regions for $L$-functions
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915235517956096 |
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| author | Wattanawanichkul, Nawapan |
| author_facet | Wattanawanichkul, Nawapan |
| contents | For integers $m, m' \ge 1$, let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(m')$, respectively. We present a new proof of zero-free regions for $L(s, π)$ and for $L(s, π\times π')$ under the assumption that $π, π'$ or $L(s,π\times π')$ is self-dual. Our approach builds on ideas of "pretentious" multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_05606 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A metric approach to zero-free regions for $L$-functions Wattanawanichkul, Nawapan Number Theory 11F66 For integers $m, m' \ge 1$, let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(m')$, respectively. We present a new proof of zero-free regions for $L(s, π)$ and for $L(s, π\times π')$ under the assumption that $π, π'$ or $L(s,π\times π')$ is self-dual. Our approach builds on ideas of "pretentious" multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi. |
| title | A metric approach to zero-free regions for $L$-functions |
| topic | Number Theory 11F66 |
| url | https://arxiv.org/abs/2504.05606 |