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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.06248 |
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| _version_ | 1866913177231425536 |
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| author | Kobayashi, Hirotaka |
| author_facet | Kobayashi, Hirotaka |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Hardy's $Z$-function and its derivatives associated with the extended Selberg class Kobayashi, Hirotaka Number Theory 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. |
| title | On Hardy's $Z$-function and its derivatives associated with the extended Selberg class |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.06248 |