Note on the number of zeros of ${\zeta }^{(k)}(s)$

Abstract

Assuming the Riemann hypothesis, we prove that $$N_k(T)=\frac{T}{2\pi }\log \frac{T}{4\pi e}+O_k\left(\frac{\log T}{\log \log T}\right),$$ where $N_k(T)$ is the number of zeros of ${\zeta }^{(k)}(s)$ in the region $0<\Im s\le T$. We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo.

10 pages