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In this paper, the approximation of analytic functions by shifts  ζ P (s + i τ)  of Beurling zeta-functions  ζ P (s)  of certain systems  P  of generalized prime numbers is discussed. It is required that the system of generalized integers  N P  generated by  P  satisfies  ∑ m ⩽ x , m ∈ N 1 = a x + O (x δ) ,  a > 0 ,  0 ⩽ δ < 1 , and the function  ζ P (s)  in some strip lying in  σ ^ < σ < 1 ,  σ ^ > δ , which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces. [ABSTRACT FROM AUTHOR]