Glasnik Matematicki, Vol. 55, No. 1 (2020), 55-65.

EXTENSION OF THE FUNCTIONAL INDEPENDENCE OF THE RIEMANN ZETA-FUNCTION

Antanas Laurinčikas

Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania
e-mail: antanas.laurincikas@mif.vu.lt

Abstract.   In 1972, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i. e., if the functions φj are continuous in ℂN and φ0(ζ(s), …, ζ(N-1)(s))+ ∙∙∙ + sn φn(ζ(s), …, ζ(N-1)(s)) ≡ 0, then φj≡ 0 for j=0,…, n. The problem goes back to Hilbert who obtained the algebraic-differential independence of ζ(s). In the paper, the functional independence of compositions F(ζ(s)) for some classes of operators F in the space of analytic functions is proved. For example, as a particular case, the functional independence of the function cosζ(s) follows.

2010 Mathematics Subject Classification.   11M06

Key words and phrases.   Algebraic-differential independence, functional independence, Riemann zeta-function, space of analytic functions, universality

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https://doi.org/10.3336/gm.55.1.05

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