Universality of the zeta function in short intervals

TL;DR

This paper advances the understanding of the Riemann zeta-function's universality by proving it in much shorter intervals, both conditionally on the Riemann Hypothesis and unconditionally, with explicit bounds.

Contribution

It establishes the universality of the zeta function in short intervals of size $( ext{log } T)^B$, improving previous results and providing explicit bounds under the Riemann Hypothesis.

Findings

Universality holds in intervals of size $( ext{log } T)^B$ assuming RH.

Unconditionally, a positive upper density of such intervals exists.

Explicit bounds for the interval length $H$ are provided.

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.