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

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 Garunkštis and Steuding. We then deduce an asymptotic of the sum $$S_T(a,\delta )=\sum _{\tau <{\gamma }_a⩽T}{\zeta }^{\mathrm{\prime }}({\rho }_a+i\delta )X^{{\rho }_a},T\to \mathrm{\infty },$$ where $0\mathrm{\ne }\delta =\frac{2\pi \alpha }{\log \frac{T}{2\pi X}}\ll 1$, and $X>0$ and $\tau ⩾\mathrm{\mid }\delta \mathrm{\mid }+1$ 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.