Original scientific paper

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

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

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.

Keywords

Algebraic-differential independence; functional independence; Riemann zeta-function; space of analytic functions; universality

Hrčak ID:

239046

URI

https://hrcak.srce.hr/239046

Publication date:

12.6.2020.

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