Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function

By Michael J. Mossinghoff, T. Trudgian. 2014

66 citations recorded by the source corpus.

Abstract We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 − 1 / ( 5.573412 log ⁡ | t | ) for | t | ≥ 2 . This represents the largest known zero-free region within the critical strip for 3.06 ⋅ 10 10 | t | exp ⁡ ( 10 151.5 ) . Our improvements result from determining some favorable trigonometric polynomials having particular properties, and from analyzing the error term in the method of Kadiri. We also improve an upper bound in a question of Landau regarding nonnegative trigonometric polynomials.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/32c31b924c60f934f2fcd35f7a28b90c3505978f

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In Algebra and Number Theory

1. Kevin O'Bryant
2. Michael A. Bennett
3. Greg Martin
4. Andrej Dujella
5. Samir Siksek
6. Clemens Fuchs
7. B. Totaro

Citations in the corpus

• The Riemann hypothesis is true up to 3·1012 (2020, unknown): “573412 by Mossinghoff and Trudgian [18].”
• Explicit bounds for primes in arithmetic progressions (2018, background): “By way of example, it provides a stronger error bound than Theorem 2 of Trudgian [43] for all 1,474,279,333 ≤ x 10621 (and sharpens corresponding results in [4] and [7] in much smaller ranges).”