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

Source: [Note on the number of zeros of $\\zeta^{(k)}(s)$](<https://www.sota2.com/research/paper/68fa587b1e209131ce9aeba7>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T15:22:37.449Z

Authors: Fan Ge, Ade Irma Suriajaya

Published: 2020-07-29T06:03:24.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2007.14617>)

[PDF](<https://arxiv.org/pdf/2007.14617.pdf>)