An effective universality theorem for the Riemann zeta function

Abstract

Let $0<r<1/4$, and $f$ be a non-vanishing continuous function in $|z|\le r$, 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 $\epsilon$, and moreover that the set of such $t\in [0,T]$ has measure at least $c(\epsilon )T$ for some $c(\epsilon )>0$, once $T$ is large enough. This was refined by Bagchi who showed that the measure of such $t\in [0,T]$ is $(c(\epsilon )+o(1))T$, for all but at most countably many $\epsilon >0$. 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.