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Keywords:

  • Riemann zeta-function;
  • universality;
  • value-distribution MSC (2010) 00-11M06

Abstract

Voronin's universality theorem claims that the Riemann zeta-function ζ can approximate any non-vanishing analytic function. We give new applications of this remarkable approximation property. In particular, we prove that the function Φ(s) ≔ F(ζ(s), ζ′(s), …, ζ(n)(s)) is universal in some sense, provided that F is sufficiently smooth (continuous or holomorphic). The proof relies on the theorem of Picard–Lindelöf and the implicit function theorem. Several examples are given.