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Differential universality


  • Dedicated to Professor Stephan Ruscheweyh on the occasion of his retirement


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.