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arXiv · 2502.15364

Universality of the zeta function in short intervals

Yoonbok Lee·Łukasz Pańkowski
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Abstract

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 $\tau\in[T,T+H]$ such that $\zeta(s+i\tau)$ approximates an arbitrary given analytic function has a positive upper density.

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Yoonbok Lee, Łukasz Pańkowski. 2025-02-21. Universality of the zeta function in short intervals. https://arxiv.org/abs/2502.15364

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