Some randomly selected arithmetical sums (Q1109815)

Let \(p_ 1(n)<p_ 2(n)<...<p_{\omega}(n)\) be the distinct prime divisors of n, where \(\omega\) (n) denotes their number. The sum \(R_ j(x)=\sum_{2\leq n\leq x}1/p_ j(n)\) has been recently investigated for some specific choices of j. In case \(j=\omega (n)\) this has been done by the reviewer [Arch. Math. 36, 57-61 (1981; Zbl 0436.10019)] and \textit{P. Erdős}, the reviewer and \textit{C. Pomerance} [Glas. Mat., III. Ser. 21(41), 283-300 (1986; Zbl 0615.10055)]. For \(k\geq 1\) fixed \textit{P. Erdős} and the reviewer [Publ. Inst. Mat., Nouv. Sér. 32(46), 49-56 (1982; Zbl 0506.10035)] showed that, as \(x\to \infty\), \[ R_{\omega - k}(x)\sim c_ k\cdot \frac{x(\log \log x)^{k-1}}{\log x}\quad (c_ k>0). \] The present authors obtain an interesting result in the probabilistic spirit by considering the sums \[ (1)\quad R(x)=\sum_{2\leq n\leq x}1/p(n), \] where for every n they pick p(n) to be one of the \(p_ j(n)'s\) with equal probabilities. The total number of sums of the form (1) is \(\omega\) (2)\(\omega\) (3)...\(\omega\) ([x]), and it is said that a property holds for almost all sums in (1) if the number N(x) of the sums with the property in question satisfies \[ \lim_{x\to \infty}N(x)/\omega (2)\omega (3)...\omega ([x])=1. \] The main result is that, for almost all sums in (1), \[ R(x)=\frac{Cx}{\log \log x}+O(\frac{x}{(\log \log x)^ 2})\quad (C=\sum_{p}1/p^ 2). \] The proof is based on a variant of Chebyshev's inequality from probability theory, which reduces the problem to the evaluation of the sum \(\sum_{p\leq x}(1/p)\sum_{m\leq x/p}(1/\omega (pm)).\) The same ideas are used to evaluate certain random sums related to the divisors of n.