Vol. 303, No. 1, 2019

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Explicit formulae and discrepancy estimates for $a$-points of the Riemann zeta-function

Siegfred Baluyot and Steven M. Gonek

Vol. 303 (2019), No. 1, 47–71
Abstract

For a fixed a0, an a-point of the Riemann zeta-function is a complex number ρa = βa + iγa such that ζ(ρa) = a. Recently J. Steuding estimated the sum

0<γaT βa>0 xρa

for a fixed x as T , and used this to prove that the ordinates γa are uniformly distributed modulo 1. We provide uniform estimates for this sum when x > 0 and 1, and T > 1. Using this, we bound the discrepancy of the sequence λγa when λ0. We also find explicit representations and bounds for the Dirichlet coefficients of the series 1(ζ(s) a) and upper bounds for the abscissa of absolute convergence of this series.

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Keywords
Riemann zeta-function, $a$-points, uniform distribution, discrepancy
Mathematical Subject Classification 2010
Primary: 11M06, 11M26
Milestones
Received: 7 January 2019
Revised: 5 May 2019
Accepted: 14 May 2019
Published: 21 December 2019
Authors
Siegfred Baluyot
Department of Mathematics
University of Rochester
Rochester, NY
United States
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States
Steven M. Gonek
Department of Mathematics
University of Rochester
Rochester, NY
United States