A note on the normal largest gap between prime factors
Résumé
Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer~$n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with $n$, we have
$$f(n):=\max_{1\leqslant j<\omega(n)}\log \Big({\log p_{j+1}(n)\over \log p_j(n)}\Big)=\log_3n+O(\xi(n))$$
for almost all integers $n$.