An effective universality theorem for the Riemann zeta function
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Volume
93
Pagination
709 - 736
Publisher
Publisher URL
DOI
10.4171/CMH/448
Journal
Commentarii Mathematici Helvetici
Issue
ISSN
1420-8946
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Show full item recordAbstract
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.
Authors
Lamzouri, Y; LESTER, SJ; Radziwill, MCollections
- Mathematics [1458]