#### 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 $a\ne 0$, 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.

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Riemann zeta-function, $a$-points, uniform distribution, discrepancy