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

DOI: 10.2140/pjm.2019.303.47

Abstract

For a fixed $a\ne 0$, an $a$-point of the Riemann zeta-function is a complex number ${\rho }_a={\beta }_a+i{\gamma }_a$ such that $\zeta \left({\rho }_a\right)=a$. Recently J. Steuding estimated the sum

$$\sum _{\begin{array}{c}0<{\gamma }_a\le T\\ {\beta }_a>0\end{array}}{x}^{{\rho }_a}$$

for a fixed $x$ as $T\to \infty$, and used this to prove that the ordinates ${\gamma }_a$ are uniformly distributed modulo $1$. We provide uniform estimates for this sum when $x>0$ and $\ne 1$, and $T>1$. Using this, we bound the discrepancy of the sequence $\lambda {\gamma }_a$ when $\lambda \ne 0$. We also find explicit representations and bounds for the Dirichlet coefficients of the series $1∕\left(\zeta \left(s\right)-a\right)$ and upper bounds for the abscissa of absolute convergence of this series.

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