An effective universality theorem for the Riemann zeta-function

TL;DR

This paper presents the first effective version of Voronin's universality theorem for the Riemann zeta function, providing explicit convergence rates and extending the approach to other L-functions and families.

Contribution

It introduces a novel method that yields an effective rate of convergence in Voronin's theorem, applicable to various L-functions and their families.

Findings

Established an explicit rate of convergence for the universality approximation.

Extended the method to other L-functions and families in different aspects.

Provided a flexible approach that improves upon previous non-effective results.

Abstract

Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $|z|\leq r$, that is analytic in the interior. Voronin's universality theorem asserts that translates of the Riemann zeta function $\zeta(3/4 + z + it)$ can approximate $f$ uniformly in $|z| < r$ to any given precision $\varepsilon$, and moreover that the set of such $t \in [0, T]$ has measure at least $c(\varepsilon) T$ for some $c(\varepsilon) > 0$, once $T$ is large enough. This was refined by Bagchi who showed that the measure of such $t \in [0,T]$ is $(c(\varepsilon) + o(1)) T$, for all but at most countably many $\varepsilon > 0$. Using a completely different approach, we obtain the first effective version of Voronin's Theorem, by showing that in the rate of convergence one can save a small power of the logarithm of $T$. Our method is flexible, and can be generalized to other $L$-functions in the $t$-aspect, as well as to families of $L$-functions in the conductor aspect.