Let P be a system of generalized prime numbers, and NP the corresponding system of generalized integers. Assuming that ∑m⩽xm∈NP1−ax≪xβ with a>0 and 0⩽β<1, we consider the Beurling zeta-function ζP(s), s=σ+it. Beurling zeta-functions constitute a wide class of non-standard zeta-functions which pose interesting mathematical problems. Numerous authors are searching for restrictions on the systems P and NP that the corresponding Beurling zeta-functions have some properties similar to those of classical zeta-functions. One of such properties is the functional independence which was initiated by O. Hölder and D. Hilbert, and, in the most general form, by S.M. Voronin. This is a motivation to obtain the functional independence in the Voronin sense for a certain class of Beurling zeta-functions. Under a certain additional condition involving the generalized von Mangoldt function, we obtain the functional independence of the function ζP(s). We prove that the function ζP(s) does not satisfy the equation ∑k=0rskFkζP(s),ζP′(s),…,ζP(n−1)(s)=0 with continuous functions Fk, k=0,…,r. The proof is based on the universality property of ζP(s) on approximation of analytic functions by shifts ζP(s+iτ), τ∈R.
Laurinčikas et al. (Thu,) studied this question.