Limit Theorem for the Dirichlet Series with Multiplicative Coefficients

Abstract

Let g(m) be a complex-valued multiplicative function such that |g(m)| ≤ 1, and let

$$Z(s) = \sum\limits_{m = 1}^\infty{\frac{{9(m)}}{{{m^2}}}} $$

(0.1)

for σ > 1. The asymptotic mean value of the function g(m), i.e.

$$\frac{1}{x}\sum\limits_{m \leqslant x} {g(m)} ,x \to \infty ,$$

(0.2)

is of great importance in the analytic number theory. The best results of such kind are obtained using the method of generating Dirichlet series, see, for example, (Halász, 1968; Levin and Timofeev, 1971; Kubilius, 1962, 1974; Elliott, 1979; Tenenbaum, 1992). This method is based on the representation of the mean value (0.2) by the contour integral of the function Z(s). For the evaluation of this integral it is necessary to know the behaviour of the function Z(s) in the neighbourhood of the line σ = 1. Thus from the functional properties of the function Z(s) the asymptotics of the mean value of its coefficients g(m) follows. It appears that an inverse relation exists, too: the functional properties of Z(s) depend on the asymptotic behaviour of the mean value (0.2). In this chapter the asymptotics of the mean value of the coefficients of the Dirichlet series are used to prove a limit theorem for the function Z(s) in the space of analytic functions. From this theorem the universality and the functional independence of Z(s) follow.