Poisson distribution for a sum of additive functions
The authors consider the Poisson law as the limit distribution of additive functions with shifted arguments. Let \(f_x\), \(g_x\) be two sets of additive functions with values \(f_x(p),\;g_x(p) \in \{0,1\}\) (at the primes). The main result ist: If the distributions \[ \nu_x\{n\leq x,\; f_x(n) + g_x(n+1) < u\} \] converge weakly (as \(x\to\infty\)) to some distribution function, then \[ \phi(x,\ell) =\frac1x \sum_{n\leq x}\prod_{r=0}^{\ell-1} (f_x(n) - g_x(n+1) - r), \] (\(\ell = 1,2,\dots\)) has a finite limit \[ \lim_{x\to\infty}\phi(x,\ell) = \phi_\ell,\tag{\(\ast\)} \] where \(\phi_\ell\) is the factorial moment of the limit law. And: if \((\ast)\) holds for every fixed \(\ell\), and if \(\sum_{\ell=1}^\infty \frac{2^\ell\phi_\ell}{\ell\,!}\) converges, then the distribution functions \(\nu_x\{\dots\}\) converge weakly to some distribution with characteristic function \[ 1 + \sum_{\ell=1}^\infty \frac1{\ell\,!} \phi_\ell (e^{it} - 1)^\ell. \] The distributions converge weakly to the Poisson law with parameter \(\lambda\), if and only if ~\(\lim_{x\to\infty}\phi(x,\ell) = \lambda^\ell\)~ for every \(\ell\). If the functions \(f_x,\; g_x\) are strongly additive, and if in addition \[ \lim_{x\to\infty} \frac1{\log x} \left( \sum_{{p\,\leq x}\atop {f_x(p)=1}} \frac{\log p}p + \sum_{{p\,\leq x}\atop {g_x(p)=1}} \frac{\log p}p \right)=0, \] then the distributions \(\nu_x(\dots)\) converge weakly to the Poisson law (with parameter \(\lambda\)) if and only if \[ \lim_{x\to\infty}\left( \max_{{p\,\leq x}\atop {f_x(p)=1}} \frac1p + \max_{{p\,\leq x}\atop {g_x(p)=1}} \frac1p \right)=0, \text{ and } \lim_{x\to\infty}\left( \sum_{{p\,\leq x}\atop {f_x(p)=1}} \frac1p + \sum_{{p\,\leq x}\atop {g_x(p)=1}} \frac1p \right)=\lambda. \]
scientific article; zbMATH DE number 5172637