Gediminas Stepanauskas

The mean values of multiplicative functions. II

as x ---> oo. There are some publications devoted to studying asymptotics of those sums [1, 2, 4, 6, 7, 9, 11, 12]. In most cases the moduli of the functions gt do not exceed 1. It is also often required in addition that the values of gt on the set of primes should be close to 1. There are results obtained when one or both of these multiplicative functions are equal to the number of divisors. But many of well-known classical functions do not fall into the above classes. It would be very interesting to know the asymptotical behavior of sums (1) for the classical Liouville or MObius functions. This is related to the unsolved problem of twins and to Goldbachs assertion that every even integer above three can be expressed as the sum of two primes (see [2]). In this paper we remove the condition that the functions Igtl are bounded by 1, but the requirement for the functions gt to be close to 1 still remains valid. Thus the results obtained here cannot be applied to the Liouville and Mrbius functions. Neither can they be applied to the enumerating function of non-isomorphic Abelian groups of finite order because the function increases too fast. In the proofs below as in [6], we follow the ideas and methods of A. Rrnyi [5], A. Hildebrand [3], and R. Warlimont [8].

THE FACTORIAL MOMENTS OF ADDITIVE FUNCTIONS WITH RATIONAL ARGUMENT

We consider the weak convergence of the set of strongly additive functions f(q) with rational argument q. It is assumed that /(p ) and /(1/p ) € {0, 1} for all primes. We obtain necessary and sufficient conditions of the convergence to the limit distribution. The proof is based on the method of factorial moments. Sieve results, and HalSszs and Ruzsas inequalities are used. We present a few examples of application of the given results to some sets of fractions.

The local behaviour of some additive functions

F will mean the gamma-funct ion. In the present paper we shall investigate the l imiting behaviour of Nx(a, f ) for some addi t ive functions f . The first result in this direction was obtained by J. Hadamard [3] and C. J. de la Vall6e-Poussin [2] as the law of prime numbers . Later many authors were in teres ted in the asymptot ic behaviour of N~(a) for the functions w and ~. Already in 1900, E. Landau [8] got an answer for fixed a. In 1953-54 L. G. Sathe [11]-[14] and A. Selberg [15] investigated N~(a) for the above ment ioned functions whenever 1 < a < c l l o g l o g x , where the positive cons tant cl depends only on the function investigated. They proved