On Hardy's Z-function and its derivatives associated with the extended Selberg class

2025/09/08 by Hirotaka Kobayashi, Kobayashi, Hirotaka
Mathematics · #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.06248

openalex publication_date 2025/09/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

Hardy's Z-function Z(t) is a real-valued function of the real variable t, and whose zeros correspond exactly to the zeros of the Riemann zeta-function on the critical line. In 2012, K. Matsuoka showed that for every non-negative integer k, there exists a T=T(k)>0 such that Z^(k+1)(t) has exactly one zero between consecutive zeros of Z^(k)(t) for t≥ T under the Riemann Hypothesis. In this paper, we extend Matsuoka's theorem to L-functions in extended Selberg class.

Citations

• On the higher derivatives of Z(t) associated with the Riemann Zeta-Function

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