Universality of the zeta function in short intervals

TL;DR

The paper advances Voronin-type universality of the Riemann zeta-function to substantially shorter intervals by showing, under the Riemann Hypothesis, universality on $[T,T+H]$ with $H=(\log T)^B$ for an explicit bound on $B$, and unconditionally proving positive upper density of $\tau$ yielding prescribed analytic approximations on the same short intervals. The key method is to approximate $\log\zeta(s+i\tau)$ by a short Dirichlet polynomial and then transfer the approximation to an arbitrary target $f(s)$ using a density argument for Dirichlet polynomials together with Kronecker–Weyl equidistribution of the phases $\tau\log p$. An unconditional variant replaces the main approximation with a sequence of $T_j$ and corresponding $H_j$, leveraging zero-density estimates to control zeros off the critical line. These results link effective short-interval approximations to zero-density phenomena and contribute to the understanding of universality in near-minimal intervals.

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

Theorems & Definitions (11)

• Theorem A: L_short
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: GS
• Lemma 2.2
• Lemma 2.3
• proof : Proof of Lemma \ref{['lem:shortDirichletpoly']}
• proof : Proof of Lemma \ref{['lem:shortDirichletpoly uncond']}
• Lemma 2.4: Lemma 1 in K-V
• Lemma 2.5: Kronecker-Weyl