The value distribution of differences of additive arithmetic functions (Q919400)

Let \(a>0\), \(A>0\), \(b\), \(B\) be integers which satisfy \(aB\ne Ab\) and \(\eta(x)\) a real-valued function defined for \(x\ge 2\). Let \(f_j\), \(j=1,2\), denote real-valued additive functions. In this paper an Erdős-Wintner type criterion is given for the convergence of the distribution functions \[ F_x(z):=[x]^{-1}\#\{1\le n\le x: f_1(an+b)-f_2(An+B)-\eta (x)\le z\}. \] The special case \(a=A=1\), \(b=1\), \(B=0\) and \(\eta(x)\equiv 0\) has been proved in a recent paper by \textit{A. Hildebrand} [Trans. Am. Math. Soc. 310, No. 1, 257--276 (1988; see the preceding review Zbl 0707.11057)]. The arguments used for the proof are mainly based on new results concerning the value distribution of complex multiplicative functions of modulus \(\le 1\).