Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917566835851264 |
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| author | Yang, Andrew |
| author_facet | Yang, Andrew |
| contents | We derive explicit upper bounds for the Riemann zeta-function $ζ(σ+ it)$ on the lines $σ= 1 - k/(2^k - 2)$ for integer $k \ge 4$. This is used to show that the zeta-function has no zeroes in the region $$σ> 1 - \frac{\log\log|t|}{21.233\log|t|},\qquad |t| \ge 3.$$ This is the largest known zero-free region for $\exp(171) \le t \le \exp(5.3\cdot 10^{5})$. Our results rely on an explicit version of the van der Corput $A^nB$ process for bounding exponential sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_03165 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region Yang, Andrew Number Theory 11M06, 11M26 (Primary) 11Y35 (Secondary) We derive explicit upper bounds for the Riemann zeta-function $ζ(σ+ it)$ on the lines $σ= 1 - k/(2^k - 2)$ for integer $k \ge 4$. This is used to show that the zeta-function has no zeroes in the region $$σ> 1 - \frac{\log\log|t|}{21.233\log|t|},\qquad |t| \ge 3.$$ This is the largest known zero-free region for $\exp(171) \le t \le \exp(5.3\cdot 10^{5})$. Our results rely on an explicit version of the van der Corput $A^nB$ process for bounding exponential sums. |
| title | Explicit bounds on $ζ(s)$ in the critical strip and a zero-free region |
| topic | Number Theory 11M06, 11M26 (Primary) 11Y35 (Secondary) |
| url | https://arxiv.org/abs/2301.03165 |