The logarithmic frequency of values of additive functions (Q1873251)

Let \(f_x\) be a set of strongly additive functions such that \(f_x(p)\in \{0,1\}\) for each prime \(p\). The objective of this paper is to establish necessary and sufficient conditions for the weak convergence of the distribution functions \[ \mu_x(f_x(m)< u)= \Biggl( \sum_{m\leq x} \frac 1m \Biggr)^{-1} \sum_{\substack{ m\leq x\\ f_x(m)=1}} \frac 1m \] to the Poisson law \(\Pi_\lambda(u)\) with positive parameter \(\lambda\). The three conditions obtained are that \[ \lim_{x\to\infty} \max_{\substack{ p\leq x\\ f_x(p)=1}} \frac 1p= 0, \quad \lim_{x\to\infty} \sum_{\substack{ p\leq x\\ f_x(p)=1}} \frac 1p= \lambda, \quad \lim_{x\to\infty} \frac{1}{\ln x} \sum_{\substack{ p\leq x\\ f_x(p)=1}} \frac{\ln p}{p}= 0. \] The proofs depend on applying well-known methods and results in probabilistic number theory. The argument for the sufficiency utilizes a lemma in [Acta Arith. 26, 333-364 (1975; Zbl 0318.10041)] by \textit{B. V. Levin} and \textit{N. M. Timofeev}, whilst for the necessity a key lemma provides an estimate of the concentration function of the distribution of an additive function with respect to the measure \(\mu_x\).