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Differential universality
Author(s) -
Christ Thomas,
Steuding Jörn,
Vlachou Vagia
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200036
Subject(s) - mathematics , universality (dynamical systems) , holomorphic function , riemann zeta function , pure mathematics , riemann hypothesis , mathematical analysis , quantum mechanics , physics
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.