# An effective universality theorem for the Riemann zeta function

### Youness Lamzouri

York University, Toronto, Canada### Stephen Lester

Queen Mary University of London, UK### Maksym Radziwiłł

McGill University, Montreal, Canada

## Abstract

Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $∣z∣≤r$, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function $ζ(3/4+z+it)$ can approximate $f$ uniformly in $∣z∣<r$ to any given precision $ε$, and moreover that the set of such $t∈[0,T]$ has measure at least $c(ε)T$ for some $c(ε)>0$, once $T$ is large enough. This was refined by Bagchi who showed that the measure of such $t∈[0,T]$ is $(c(ε)+o(1))T$, for all but at most countably many $ε>0$. 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.

## Cite this article

Youness Lamzouri, Stephen Lester, Maksym Radziwiłł, An effective universality theorem for the Riemann zeta function. Comment. Math. Helv. 93 (2018), no. 4, pp. 709–736

DOI 10.4171/CMH/448