TY - JOUR

T1 - An effective universality theorem for the Riemann zeta function

AU - Lamzouri, Youness

AU - Lester, Stephen

AU - Radziwill, Maksym

N1 - Publisher Copyright:
© Swiss Mathematical Society.

PY - 2018

Y1 - 2018

N2 - Let 0 < r < 1=4, and f be a non-vanishing continuous function in jzj r, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function .3=4 C z C can approximate f uniformly in jzj < r to any given precision ", and moreover that the set of such t 2 Œ0; T has measure at least c."T for some c." > 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t 2 Œ0; T is .c."C o.1T, for all but at most countably many " > 0. Using a completely di erent approach, we obtain the first e ective 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.

AB - Let 0 < r < 1=4, and f be a non-vanishing continuous function in jzj r, that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function .3=4 C z C can approximate f uniformly in jzj < r to any given precision ", and moreover that the set of such t 2 Œ0; T has measure at least c."T for some c." > 0, once T is large enough. This was refined by Bagchi who showed that the measure of such t 2 Œ0; T is .c."C o.1T, for all but at most countably many " > 0. Using a completely di erent approach, we obtain the first e ective 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.

KW - Riemann zeta function

KW - Universality

UR - http://www.scopus.com/inward/record.url?scp=85057984020&partnerID=8YFLogxK

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U2 - 10.4171/CMH/448

DO - 10.4171/CMH/448

M3 - Article

AN - SCOPUS:85057984020

SN - 0010-2571

VL - 93

SP - 709

EP - 736

JO - Commentarii Mathematici Helvetici

JF - Commentarii Mathematici Helvetici

IS - 4

ER -