The distribution of the number of factors in a factorization
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A factorization of a positive integer n, here, is a specification of m(d), the power to which d occurs in Π dm(d) = n; order is immaterial. The number of factors in a factorization has two natural interpretations: as Σ m(d) or as the number of nonzero m(d), that is, counting or not counting multiplicity. In either case, the factorizations of positive integers ≤ x into k factors number approximately ψv(u) × (log x)k - 1 k!(k - 1)!, where u = k(k - 1) log x, and ψv is either Γ(2 - u) or 1 Γ(1 + u) according to whether multiplicity is counted or not. In the former case, we must have u ≤ 2 - ε; in the latter, u ≤ C. © 1987.
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