On the abscissae of Weil representation zeta functions for procyclic groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917859424206848 |
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| author | Kionke, Steffen |
| author_facet | Kionke, Steffen |
| contents | A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group $G$ only depends on the set $S$ of primes dividing the order of $G$ and that it agrees with the abscissa of the Dedekind zeta function of $\mathbb{Z}[p^{-1}\mid p \not\in S]$. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number $1 \leq β\leq 2$ is the Weil abscissa of some procyclic group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12848 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the abscissae of Weil representation zeta functions for procyclic groups Kionke, Steffen Group Theory Number Theory 20P05, 20E18, 11M41 A famous conjecture of Chowla on the least primes in arithmetic progressions implies that the abscissa of convergence of the Weil representation zeta function for a procyclic group $G$ only depends on the set $S$ of primes dividing the order of $G$ and that it agrees with the abscissa of the Dedekind zeta function of $\mathbb{Z}[p^{-1}\mid p \not\in S]$. Here we show that these consequences hold unconditionally for random procyclic groups in a suitable model. As a corollary, every real number $1 \leq β\leq 2$ is the Weil abscissa of some procyclic group. |
| title | On the abscissae of Weil representation zeta functions for procyclic groups |
| topic | Group Theory Number Theory 20P05, 20E18, 11M41 |
| url | https://arxiv.org/abs/2411.12848 |