# Commentarii Mathematici Helvetici

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**Volume 93, Issue 4, 2018, pp. 709–736**

**DOI: 10.4171/CMH/448**

Published online: 2018-11-20

An effective universality theorem for the Riemann zeta function

Youness Lamzouri^{[1]}, Stephen Lester

^{[2]}and Maksym Radziwiłł

^{[3]}(1) York University, Toronto, Canada

(2) Queen Mary University of London, UK

(3) McGill University, Montreal, Canada

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.

*Keywords: *Riemann zeta function, universality

Lamzouri Youness, Lester Stephen, Radziwiłł Maksym: An effective universality theorem for the Riemann zeta function. *Comment. Math. Helv.* 93 (2018), 709-736. doi: 10.4171/CMH/448