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

TL;DR

Under the assumption of the Riemann hypothesis, the paper derives an asymptotic formula for the number of zeros of the k-th derivative of the Riemann zeta function and extends the method to Selberg zeta functions.

Contribution

The paper provides a new zero counting formula for derivatives of the Riemann zeta function and applies the technique to Selberg zeta functions, improving previous results.

Findings

Asymptotic zero count for th derivatives of (zeta) under RH

Zero counting formula for derivatives of Selberg zeta functions

Improvement over earlier work of Luo on Selberg zeta zeros

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.