A note on the normal largest gap between prime factors

TL;DR

This paper investigates the behavior of the largest gap between the logarithms of consecutive prime factors of integers, showing that for almost all integers, this gap is approximately \\log_3 n with small fluctuations.

Contribution

It provides a detailed proof of Erd\

Findings

The maximum gap between logs of consecutive prime factors is approximately \\log_3 n for almost all integers.

The result holds with small fluctuations characterized by any function \\xi(n) tending to infinity.

The paper refines understanding of the distribution of prime factors within integers.

Abstract

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 $n$.