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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.07696 |
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| _version_ | 1866916648521302016 |
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| author | Stopple, Jeffrey |
| author_facet | Stopple, Jeffrey |
| contents | Study of the level curve for the real part of $η(s)=0$ with $η(s)=π^{-s/2}Γ(s/2)ζ^\prime(s)$ gives a new classification of the zeros of $ζ(s)$ and of $ζ^\prime(s)$. We conjecture that for type 2 zeros, $\liminf (β^\prime -1/2)\logγ^\prime = 0$ if and only if $\liminf (γ^+-γ^-)\log γ^\prime=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify $10^6$ zeros of $ζ^\prime(s)$ near $T=10^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Level curves for Zhang's Eta Function Stopple, Jeffrey Number Theory 11M06, 15B52, 30C15 Study of the level curve for the real part of $η(s)=0$ with $η(s)=π^{-s/2}Γ(s/2)ζ^\prime(s)$ gives a new classification of the zeros of $ζ(s)$ and of $ζ^\prime(s)$. We conjecture that for type 2 zeros, $\liminf (β^\prime -1/2)\logγ^\prime = 0$ if and only if $\liminf (γ^+-γ^-)\log γ^\prime=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify $10^6$ zeros of $ζ^\prime(s)$ near $T=10^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices. |
| title | Level curves for Zhang's Eta Function |
| topic | Number Theory 11M06, 15B52, 30C15 |
| url | https://arxiv.org/abs/2503.07696 |