Work

Universality of the zeta function in short intervals

Year: 2025

Type: article

Abstract: We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $$[T,T+H]$$ [ T , T + H ] . Assuming the Riemann Hypothesis, we prove that universality in such short intervals holds for $$H=(\log T)^B$$ H = ( log T ) B with an explicitly given $$B>0$$ B > 0 . Unconditionally, we show that for the same H the set of real numbers $$\tau \in [T,T+H]$$ τ ∈ [ T , T + H ] such that $$\zeta (s+i\tau )$$ ζ ( s + i τ ) approximates an arbitrary given analytic function has a positive upper density.

Source: The Ramanujan Journal

DOI: 10.1007/s11139-025-01144-0

Authors: Yoonbok Lee, Łukasz Pańkowski

Institutions: Incheon National University, Adam Mickiewicz University in Poznań

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990