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arXiv · 2509.06248

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

Hirotaka Kobayashi
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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 $t\ge T$ under the Riemann Hypothesis. In this paper, we extend Matsuoka's theorem to $L$-functions in extended Selberg class.

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Hirotaka Kobayashi. 2025-09-08. On Hardy's $Z$-function and its derivatives associated with the extended Selberg class. https://arxiv.org/abs/2509.06248

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