Explicit estimates for the logarithmic derivative and the reciprocal of the Riemann zeta function

2024/05/08 by Leong, Nicol
#11M06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.04869

Abstract

In this article, we give explicit bounds of order log t for σ close to 1, for two quantities: |ζ'(σ+it)/ζ(σ+it)| and |1/ζ(σ+it)|. We correct an error in the literature, and especially in the case of |1/ζ(σ+it)|, also provide improvements in the constants. Using an argument involving the trigonometric polynomial, we additionally provide a slight asymptotic improvement within the classical zero-free region: 1/ζ(σ+it) ≪ (log t)^11/12. The same method applied to the Korobov--Vinogradov zero-free region gives a new record: the unconditional bound 1/ζ(σ+it) ≪ (log t)^2/3(loglog t)^1/4.

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