Treffer: The asymptotic distribution of factorizations of natural numbers into prime divisors

For each natural number n, writing its prime factorization as \(n=p_ 1... p_ r\), \(p_ 1\geq p_ 2\geq...\geq p_ r\), let \(p_ i=p_ i(n)=n^{x_ i}\). The author announces several results on the distribution of these exponents \(x_ i\). A typical result: For all \(k=1,2,..\). and \(\alpha_ 1,...,\alpha_ k\in [0,1]\) one has, letting \(x_ m(n) : x_{m-1}(n)=q_ m(n)\), \[ \lim_{m\to \infty}\lim_{N\to \infty}(1/N) \#\{n\leq N: q_{m+i}(n)\leq \alpha_ i,\quad i=1,...,k\}=\alpha_ 1... \alpha_ k\quad, \] i.e., the ratios of the logarithms of neighboring prime divisors are asymptotically independent and uniformly distributed.