On the distribution of values of the derivative of the Riemann zeta function at its zeros.~I

Akio Fujii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 57-82

Résumé

Let $\zeta'(s)$ be the derivative of the Riemann zeta function $\zeta(s)$. A study on the value distribution of $\zeta'(s)$ at the non-trivial zeros $\rho$ of $\zeta(s)$ is presented. In particular, for a fixed positive number $X$, an asymptotic formula and a non-trivial upper bound for the sum $\sum_{0\operatorname{Im}\rho\leq T}\zeta'(\rho)X^\rho$ as $T\to\infty$ are given. We clarify the dependence on the arithmetic nature of $X$.