Explicit bounds for $\zeta$ and a new zero free region

Speaker: Chiara Bellotti

Date: Tue, Jan 16, 2024

Location: PIMS, University of Lethbridge, Online, Zoom

Conference: Analytic Aspects of L-functions and Applications to Number Theory

Subject: Mathematics, Number Theory

Class: Scientific

CRG: L-Functions in Analytic Number Theory


In this talk, we prove that |ζ(σ+it)|≤ 70.7 |t|4.438(1-σ)^{3/2} log2/3|t| for 1/2≤ σ ≤ 1 and |t| ≥ 3, combining new explicit bounds for the Vinogradov integral with exponential sum estimates. As a consequence, we improve the explicit zero-free region for ζ(s), showing that ζ(σ+it) has no zeros in the region σ ≥ 1-1/(53.989 (log|t|)2/3(log log|t|)1/3) for |t| ≥ 3.

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