The Number of Zeros of $\zeta'(s)$

TL;DR

Under the assumption of the Riemann Hypothesis, the paper derives an asymptotic formula for counting the zeros of the derivative of the Riemann zeta function, providing insight into their distribution.

Contribution

The paper establishes a precise asymptotic count for the zeros of ta'(s) assuming the Riemann Hypothesis, advancing understanding of their distribution.

Findings

Asymptotic formula for the number of zeros of ta'(s)

Zero count matches the main term rac{T}{2\u03c0}\u2212log rac{T}{4\u03c0}e

Error term is bounded by rac{( T)}{     T} for some slowly growing function.

Abstract

Assuming the Riemann Hypothesis, we prove that $$ N_1(T) = \frac{T}{2\pi}\log \frac{T}{4\pi e} + O\bigg(\frac{\log T}{\log\log T}\bigg), $$ where $N_1(T)$ is the number of zeros of $\zeta'(s)$ in the region $0<\Im s\le T$.