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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.01512 |
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| _version_ | 1866914781549559808 |
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| author | Sheth, Arshay |
| author_facet | Sheth, Arshay |
| contents | Let $π$ be an irreducible cuspidal automorphic representation of $\text{GL}_n(\mathbb A_\mathbb Q)$ with associated $L$-function $L(s, π)$. We study the behaviour of the partial Euler product of $L(s, π)$ at the center of the critical strip. Under the assumption of the Generalized Riemann Hypothesis for $L(s, π)$ and assuming the Ramanujan--Petersson conjecture when necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point that confirms a conjecture of Kurokawa. As an application, we obtain results towards Chebyshev's bias in the recently proposed framework of Aoki-Koyama. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Euler Products at the Centre and Applications to Chebyshev's Bias Sheth, Arshay Number Theory Let $π$ be an irreducible cuspidal automorphic representation of $\text{GL}_n(\mathbb A_\mathbb Q)$ with associated $L$-function $L(s, π)$. We study the behaviour of the partial Euler product of $L(s, π)$ at the center of the critical strip. Under the assumption of the Generalized Riemann Hypothesis for $L(s, π)$ and assuming the Ramanujan--Petersson conjecture when necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point that confirms a conjecture of Kurokawa. As an application, we obtain results towards Chebyshev's bias in the recently proposed framework of Aoki-Koyama. |
| title | Euler Products at the Centre and Applications to Chebyshev's Bias |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.01512 |