Vol. 303, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
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

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.

PDF Access Denied

However, your active subscription may be available on Project Euclid at

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Riemann zeta-function, $a$-points, uniform distribution, discrepancy
Mathematical Subject Classification 2010
Primary: 11M06, 11M26
Received: 7 January 2019
Revised: 5 May 2019
Accepted: 14 May 2019
Published: 21 December 2019
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