Explicit zero-free regions for the Riemann zeta-function
Abstract
We prove that the Riemann zeta-function $\zeta(\sigma + it)$ has no zeros in the region $\sigma \geq 1 - 1/(55.241(\log|t|)^{2/3} (\log\log |t|)^{1/3})$ for $|t|\geq 3$. In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region $\sigma \geq 1 - 1/(5.558691\log|t|)$ for $|t|\geq 2$. We also provide new bounds that are useful for intermediate values of $|t|$. Combined, our results improve the largest known zero-free region within the critical strip for $3\cdot10^{12} \leq |t|\leq \exp(64.1)$ and $|t| \geq \exp(1000)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2022
- DOI:
- 10.48550/arXiv.2212.06867
- arXiv:
- arXiv:2212.06867
- Bibcode:
- 2022arXiv221206867M
- Keywords:
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- Mathematics - Number Theory;
- 11M26;
- 42A05;
- 11Y35
- E-Print:
- 27 pages