Mean behaviour and distribution properties of multiplicative functions

Mean-value theorems with sharp quantitative remainder term estimates and theorems on the distribution of values are proved for a large class of multiplicative functions. This class is characterized by the existence of an abscissa of absolute convergence ≤ 1 for the Dirichlet series of the quotient , where is the Dirichlet series associated with ƒ and ζ(s) denotes the Riemann zeta function. In particular, the mean behaviour of ƒ on certain sequences a, namely, that of Mersenne numbers 2n −1 and that of shifted primes p−1 with natural n and prime p, respectively, is studied and cluster points of the image sequence f(a) of a are determined. The strength of the results is demonstrated by new results for some specific functions.