An effective universality theorem for the Riemann zeta function
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Abstract
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.
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