| Abstract [eng] |
Let π« be a system of generalized prime numbers, and π©π« the corresponding system of generalized integers. Assuming that βπβ©½π₯πβπ©π«1βππ₯βͺπ₯π½ with π>0 and 0β©½π½<1 , we consider the Beurling zeta-function ππ«(π ) , π =π+ππ‘. 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 π« and π©π« 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. HΓΆlder and D. Hilbert, and, in the most general form, by S.M. Voronin. This is a motivation to obtain the unctional 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 ππ«(π ). We prove that the function ππ«(π ) does not satisfy the equation βπ=0ππ ππΉπ(ππ«(π ),πβ²π«(π ),β¦,π(πβ1)π«(π ))=0 with continuous functions πΉπ, π=0,β¦,π. The proof is based on the universality property of ππ«(π ) on approximation of analytic functions by shifts ππ«(π +ππ) , πββ. |
