The distribution of the number of factors in a factorization
The author considers as a factorization of an integer n a mapping \(m: \{2,3,4,...\}\to \{0,1,2,...\}\) such that \(\prod^{\infty}_{d=2}d^{m(d)}=n\). The integer n of which m is a factorization is denoted by \(| m|\). Factorizations of this type have been studied by \textit{A. Oppenheim} [J. Lond. Math. Soc. 2, 123-130 (1927)], who obtained an asymptotic formula for the quantity {\#}\(\{\) \(m: | m| \leq x\}.\) In the paper under review, the author defines, in analogy to the familiar arithmetic functions \(\Omega\) and \(\omega\) that count the number of prime factors, the number of factors in a factorization m counted with or without multiplicity by \(\Omega (m)=\sum^{\infty}_{d=2}m(d)\) and \(\omega (m)=\#\{d: m(d)>0\}\), and introduces the counting functions \[ N_{\nu}(x,k)=\#\{m: | m| \leq x\quad and\quad \nu (m)=k\}, \] where \(\nu\) denotes either \(\Omega\) or \(\omega\). His main result is the estimate \[ N_{\Omega}(x,k)=\frac{\Gamma (2-u) x (\log x)^{k-1}}{k! (k-1)!}(1+O_{\epsilon}(\sqrt{k/\log x})) \] with \(u=k(k-1)/\log x\), which is valid uniformly for \(u\leq 2-\epsilon\) for any fixed \(\epsilon >0\), and a similar estimate for \(N_{\omega}(x,k)\). The proof is analytic and follows \textit{A. Selberg}'s approach in the corresponding problem for factorizations into prime factors [J. Indian Math. Soc., New. Ser. 18, 83-87 (1954; Zbl 0057.285)].
scientific article; zbMATH DE number 4000045
scientific article