Notes on Universality in Short Intervals and Exponential Shifts

TL;DR

This article advances Voronin-type universality for the Riemann zeta-function in short intervals by sharpening the admissible interval length from $H\ge T^{1/3+\varepsilon}$ to $H\ge T^{1273/4053}$ via Bourgain–Watt mean-square bounds on the critical line. It also extends universality to shifts that grow exponentially, through a framework allowing exponential-type shifts and a general class $\Phi$ of admissible $\phi$, preserving universality. The authors further derive restricted universality results using optimized exponent pairs, and establish conditional results under the Lindelöf hypothesis, and under the Riemann hypothesis, even shorter $H$ are possible. Collectively, these findings show the universality phenomenon for $\zeta(s)$ remains robust under shorter intervals and broad families of vertical shifts, with implications for the distribution of zeta-values in short ranges and potential extensions to other $L$-functions.

Abstract

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurin\v cikas in a problem session of a recent workshop on universality.

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Theorems & Definitions (10)

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