An effective universality theorem for the Riemann zeta function
Commentarii Mathematici Helvetici
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Let [formula], and f be a non-vanishing continuous function in [formula], that is analytic in the interior. Voronin's universality theorem asserts that translates of the Riemann zeta function [formula] can approximate f uniformly in [formula] to any given precision ε, and moreover that the set of such [formula] has measure at least [formula] for some [formula], once T is large enough. This was refined by Bagchi who showed that the measure of such t∈[0,T] is [formula] T, for all but at most countably many [formula]. 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.
AuthorsLamzouri, Y; LESTER, SJ; Radziwill, M
- College Publications