For any \(a\in {\mathbb {C}}\) , the zeros of \(\zeta (s)-a\) , denoted by \(\rho _a=\beta _a+\textrm{i}\gamma _a\) , are called the 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 expansion of the sum \(\begin{aligned} S_T(a,\delta )=\sum _{\tau <\gamma _a\leqslant T}\zeta '(\rho _a+\textrm{i}\delta )X^{\rho _a},\quad T\rightarrow \infty , \end{aligned}\) where \(X>0\) and \(\tau \geqslant |\delta |+1\) are fixed, and \(0\ne \delta =\frac{2\pi \alpha }{\log \frac{T}{2\pi X}}\ll 1\) . We also find the interesting varied behavior of \(S_T(a,\delta )\) in different X ranges, which is more complicated than those described previously by Gonek and Pearce-Crump.
