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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
Abstract

We prove lower bounds for the discrete negative 2k-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.

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