Explicit zero-free regions for the Riemann zeta-function

2022/12/13 by Mossinghoff, Michael J., Trudgian, Timothy S., Yang, Andrew · 4 citations
#11M26 #11Y35 #42A05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2212.06867

Abstract

We prove that the Riemann zeta-function ζ(σ+ it) has no zeros in the region σ≥ 1 - 1/(55.241(log|t|)^2/3 (loglog |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/(5.558691log|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⋅10¹² ≤ |t|≤ exp(64.1) and |t| ≥ exp(1000).

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• Unconditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros
• Effective Upper Bound Estimates for |ζ'(1/2+it)| via Exponential Sums
• Transcendency of variants of Mills' constant
• New zero-free regions for Dedekind zeta-functions at small and large ordinates

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