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

Abstract

For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$ shown by Garunk\v{s}tis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,\delta)=\sum_{\tau<\gamma_a\leqslant T}\zeta'(\rho_a+i\delta)X^{\rho_a},\quad T\to\infty,\] where $0\ne\delta=\frac{2\pi\alpha}{\log\frac{T}{2\pi X}}\ll 1$, and $X>0$ and $\tau\geqslant|\delta|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,\delta)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump.