Note on the number of zeros of ζ ^(k)(s)

2020/03/17 by Fan Ge, Ade Irma Suriajaya
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics #Geometry #Mathematical analysis #Mathematics #Number theory #Pi #Riemann hypothesis #Riemann zeta function #Zero (linguistics) #math.NT #msc:11M06

paper · pdf · doi:10.1007/s11139-019-00219-z

published in The Ramanujan Journal 55(2), 661-672 (Springer Science+Business Media) · 10 pages

openalex publication_date 2020/03/17 · arxiv created 2020/07/29 · arxiv updated 2021/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Assuming the Riemann hypothesis, we prove that N_k(T) = (T)/(2π)log (T)/(4πe) + O_k(\fraclogTloglogT), where N_k(T) is the number of zeros of ζ^(k)(s) in the region 0<\Im s≤ T. We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo.

Citations

• Zeros of the derivatives of the Riemann zeta-function