An effective universality theorem for the Riemann zeta function

  • Youness Lamzouri

    York University, Toronto, Canada
  • Stephen Lester

    Queen Mary University of London, UK
  • Maksym Radziwiłł

    McGill University, Montreal, Canada

Abstract

Let , and be a non-vanishing continuous function in , that is analytic in the interior. Voronin’s universality theorem asserts that translates of the Riemann zeta function can approximate uniformly in to any given precision , and moreover that the set of such has measure at least for some , once is large enough. This was refined by Bagchi who showed that the measure of such is , for all but at most countably many . 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 . Our method is flexible, and can be generalized to other -functions in the -aspect, as well as to families of -functions in the conductor aspect.

Cite this article

Youness Lamzouri, Stephen Lester, Maksym Radziwiłł, An effective universality theorem for the Riemann zeta function. Comment. Math. Helv. 93 (2018), no. 4, pp. 709–736

DOI 10.4171/CMH/448