An effective universality theorem for the Riemann zeta function

Let 0 < r < 1=4, and f be a non-vanishing continuous function in jzj r, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function .3=4 C z C can approximate f uniformly in jzj < r to any given precision ", and moreover that the set of such t 2 Œ0; T has measure at least c."T for some c." > 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t 2 Œ0; T is .c."C o.1T, for all but at most countably many " > 0. Using a completely di erent approach, we obtain the first e ective 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.