Explicit zero-free regions for the Riemann zeta-function

We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 - 1 / ( 55.241 ( log | t | ) 2 / 3 ( log log | t | ) 1 / 3 ) for | t | ≥ 3 . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region σ ≥ 1...

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Bibliographic Details
Published in:Research in number theory 2024-03, Vol.10 (1), Article 11
Main Authors: Mossinghoff, Michael J., Trudgian, Timothy S., Yang, Andrew
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Number Theory
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Summary:We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 - 1 / ( 55.241 ( log | t | ) 2 / 3 ( log log | t | ) 1 / 3 ) for | t | ≥ 3 . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region σ ≥ 1 - 1 / ( 5.558691 log | t | ) for | t | ≥ 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 · 10 12 ≤ | t | ≤ exp ( 64.1 ) and | t | ≥ exp ( 1000 ) .
ISSN:2522-0160
2363-9555
DOI:10.1007/s40993-023-00498-y