Lower bounds for discrete negative moments of the Riemann zeta function

Heap, Winston; Li, Junxian; Zhao, Jing

doi:10.2140/ant.2022.16.1589

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Lower bounds for discrete negative moments of the Riemann zeta function

Winston Heap, Junxian Li and Jing Zhao

Vol. 16 (2022), No. 7, 1589–1625

DOI: 10.2140/ant.2022.16.1589

Abstract

We prove lower bounds for the discrete negative $2 k$ -th moment of the derivative of the Riemann zeta function for all fractional $k$ . The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.

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Keywords

Riemann zeta-function, twisted discrete moment, Gonek's conjecture

Mathematical Subject Classification

Primary: 11M06, 11N99

Milestones

Received: 21 April 2020

Revised: 29 July 2021

Accepted: 10 October 2021

Published: 16 October 2022

Authors

Winston Heap
Department of Mathematics Shandong University Jinan China

Junxian Li
University of Bonn Bonn Germany

Jing Zhao
Max Planck Institute for Mathematics Bonn Germany

Vol. 16, No. 7, 2022