The value distribution of differences of additive arithmetic functions
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 \(\le 1\).
convergence of the distribution functions
scientific article; zbMATH DE number 4160866
scientific article