Glasnik matematički, Vol. 55 No. 1, 2020.
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
Publication date:
12.6.2020.
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