Extension of the functional independence of the Riemann zeta-function

Abstract

In 1972, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i. e., if the functions Φj are continuous in C and Φ0(ζ(s), . . . , ζ(N−1)(s)) + · · ·+ sΦn(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 cos ζ(s) follows.