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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) http://resolver.caltech.edu/CaltechAUTHORS:20180612-153643797

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Abstract

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)
Related URLs:
URLURL TypeDescription
https://arxiv.org/abs/1611.10325arXivDiscussion Paper
Additional Information:The first and third authors are partially supported by Discovery Grants from the Natural Sciences and Engineering Research Council of Canada.
Funders:
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
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20180612-153643797
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:87033
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:12 Jun 2018 22:40
Last Modified:12 Jun 2018 22:40

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