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

Abstract

For any $a\in \mathbb{C}$, the zeros of $\hat{I}\P (s)-a$, denoted by $\ddot{I}{}_a={\hat{I}}_a^2+i{\hat{I}}_a^3$, are called $a$-points of the Riemann zeta function $\hat{I}\P (s)$. In this paper, we reformulate some basic results about the $a$-points of $\hat{I}\P (s)$ shown by GarunkÅ¡tis 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 $0\ne \hat{I}´=\frac{2\ddot{I}€\hat{I}\pm }{\log \frac{T}{2\ddot{I}€X}}\ll 1$, and $X>0$ and $\ddot{I}„⩾|\hat{I}´|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,\hat{I}´)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump.

16 pages, 0 figures