Note on the $a$-points of the Riemann zeta function

Abstract

For any $a\in\mathbb{C}$, the zeros of $ζ(s)-a$, denoted by $ρ_a=β_a+iγ_a$, are called $a$-points of the Riemann zeta function $ζ(s)$. In this paper, we reformulate some basic results about the $a$-points of $ζ(s)$ shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,δ)=\sum_{τ<γ_a\leqslant T}ζ'(ρ_a+iδ)X^{ρ_a},\quad T\to\infty,\] where $0\neδ=\frac{2πα}{\log\frac{T}{2πX}}\ll 1$, and $X>0$ and $τ\geqslant|δ|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,δ)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump.