Universality of the zeta function in short intervals
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866929724159164416 |
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| author | Lee, Yoonbok Pańkowski, Łukasz |
| author_facet | Lee, Yoonbok Pańkowski, Łukasz |
| contents | We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short intervals holds for $H=(\log T)^B$ with an explicitly given $B>0$. Unconditionally, we show that for the same $H$ the set of real numbers $τ\in[T,T+H]$ such that $ζ(s+iτ)$ approximates an arbitrary given analytic function has a positive upper density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universality of the zeta function in short intervals Lee, Yoonbok Pańkowski, Łukasz Number Theory 11M41 We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short intervals holds for $H=(\log T)^B$ with an explicitly given $B>0$. Unconditionally, we show that for the same $H$ the set of real numbers $τ\in[T,T+H]$ such that $ζ(s+iτ)$ approximates an arbitrary given analytic function has a positive upper density. |
| title | Universality of the zeta function in short intervals |
| topic | Number Theory 11M41 |
| url | https://arxiv.org/abs/2502.15364 |