On the ratio between two factorization functions (Q6581193)

Let \(f(n)\) denote the number of factorizations of the positive integer \(n\), i.e., the number of ways \(n\) can be expressed as an unordered product of integers greater than \(1.\) Similarly, let \(F(n)\) represents the number of factorizations of \(n\) into distinct parts. The author in an older paper proved the following result.\N\N\textbf{Theorem 1}. Let \( \Omega(n) \) be the number of prime factors of \( n \) counted with multiplicity. Then:\N\[\N\frac{f(n)}{F(n)} \leq 2^{\Omega(n)/2}.\N\]\NIn the present paper under review, the author provides an improvement of the previous Theorem.\N\N\textbf{Theorem 2}. As \( n \to \infty \), we have:\N\[\Nf(n) = F(n) \cdot \exp\left(O\left(\frac{\log n}{\log \log n}\right)\right).\N\]\NIf \(\Omega(n)\) is large, Theorem 2 is an improvement of Theorem 1.