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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2411.08364 |
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| _version_ | 1866929639320977408 |
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| author | Roy, Arindam You, Kevin |
| author_facet | Roy, Arindam You, Kevin |
| contents | Consider the approximation $\tilde{Z}_N(s) = \sum_{n=1}^N n^{-s} + χ(s) \sum_{n=1}^N n^{1-s}$ of the Riemann zeta function $ζ(s)$,
where $χ(s)$ is the ratio of the gamma functions.
This arise from the approximate functional equation of $ζ(s)$.
Gonek and Montgomery have shown that $\tilde{Z}_N(s)$ has 100\% of its zeros lie on the critical line.
Recently, $a$-values of $\tilde{Z}_N(s)$ for non-zero complex number $a$ are studied and it has been shown that the $a$-values of $\tilde{Z}_N(s)$ are cluster arbitrarily close to the critical line.
In this paper, we show that, despite the above, 0\% of non-zero $a$-values of $\tilde{Z}_N(s)$ actually lie on the critical line itself. For $ζ(s)$ at most $50\%$ non-zero $a$-values lie on the critical line is known due to Lester. We also extend our results to approximations of a wider class of $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-zero values of a family of approximations of a class of $L$-functions Roy, Arindam You, Kevin Number Theory Consider the approximation $\tilde{Z}_N(s) = \sum_{n=1}^N n^{-s} + χ(s) \sum_{n=1}^N n^{1-s}$ of the Riemann zeta function $ζ(s)$, where $χ(s)$ is the ratio of the gamma functions. This arise from the approximate functional equation of $ζ(s)$. Gonek and Montgomery have shown that $\tilde{Z}_N(s)$ has 100\% of its zeros lie on the critical line. Recently, $a$-values of $\tilde{Z}_N(s)$ for non-zero complex number $a$ are studied and it has been shown that the $a$-values of $\tilde{Z}_N(s)$ are cluster arbitrarily close to the critical line. In this paper, we show that, despite the above, 0\% of non-zero $a$-values of $\tilde{Z}_N(s)$ actually lie on the critical line itself. For $ζ(s)$ at most $50\%$ non-zero $a$-values lie on the critical line is known due to Lester. We also extend our results to approximations of a wider class of $L$-functions. |
| title | Non-zero values of a family of approximations of a class of $L$-functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.08364 |