An effective universality theorem for the Riemann zeta function

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.