Tauberian theorems and mean values of multiplicative functions

One gives an application of Tauberian theorems to the problem of connection between the mean values of multiplicative functions for the cases when the argument runs through all natural numbers and all prime numbers, respectively.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 91, pp. 145–157, 1979.The author dedicates this paper to A. I. Vinogradov on the occasion of his 50th birthday.