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

Michael J. Mossinghoff; Timothy S. Trudgian

doi:10.1016/j.jnt.2015.05.010

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

Michael J. Mossinghoff, Timothy S. Trudgian^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

37 Citations (Scopus)

Abstract

We prove that the Riemann zeta-function ζ(σ. +. it) has no zeros in the region σ≥1-1/(5.573412logL{cyrillic}|t|) for |t|≥2. This represents the largest known zero-free region within the critical strip for 3.06{dot operator}1010<|t|<expL{cyrillic}(10151.5). 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.

Original language	English
Pages (from-to)	329-349
Number of pages	21
Journal	Journal of Number Theory
Volume	157
DOIs	https://doi.org/10.1016/j.jnt.2015.05.010
Publication status	Published - 1 Dec 2015

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abstract = "We prove that the Riemann zeta-function ζ(σ. +. it) has no zeros in the region σ≥1-1/(5.573412logL{cyrillic}|t|) for |t|≥2. This represents the largest known zero-free region within the critical strip for 3.06{dot operator}1010<|t|",

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JO - Journal of Number Theory

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

Abstract

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