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An effective universality theorem for the Riemann zeta-function

Lamzouri, Youness and Lester, Stephen and Radziwiłł, Maksym (2016) An effective universality theorem for the Riemann zeta-function. . (Submitted)

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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.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Lester, Stephen0000-0003-1977-9205
Additional Information:The first and third authors are partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Classification Code:2010 Mathematics Subject Classification: Primary 11M06
Record Number:CaltechAUTHORS:20180612-153643797
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87033
Deposited By: Tony Diaz
Deposited On:12 Jun 2018 22:40
Last Modified:05 Nov 2019 17:32

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