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

Abstract

Assuming the Riemann hypothesis, we prove that

$$\begin{aligned} N_k(T) = \frac{T}{2\pi }\log \frac{T}{4\pi e} + O_k\bigg (\frac{\log T}{\log \log T}\bigg ), \end{aligned}$$

where \(N_k(T)\) is the number of zeros of \(\zeta ^{(k)}(s)\) in the region \(0<\mathfrak {I}s\le T\). We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo (Am J Math 127(5):1141–1151, 2005).

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Acknowledgements

This project was started when the first author was a postdoc fellow at the University of Waterloo and the second author was a member of iTHEMS under RIKEN Special Postdoctoral Researcher program.

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Correspondence to Fan Ge.

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Ade Irma Suriajaya is supported by JSPS KAKENHI Grant Number 18K13400.

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Ge, F., Suriajaya, A.I. Note on the number of zeros of \(\zeta ^{(k)}(s)\). Ramanujan J 55, 661–672 (2021). https://doi.org/10.1007/s11139-019-00219-z

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Keywords

  • Riemann zeta function
  • Derivatives of Riemann zeta function
  • Riemann Hypothesis

Mathematics Subject Classification

  • 11M06