Let $k$ be a fixed positive integer or even positive real number. Let $n$ be a large integer. I am interested in the following question:
What is the probability that the largest prime power divisor of $n$ is smaller than $(\log(n))^k$?
Surely this probabilty goes to $0$ when $n$ goes to infinity, but I wonder how fast it goes to $0$. For example, by the prime number theorem, every integer $n$ has a prime power divisor bigger than $(1 - \epsilon)\log(n)$, so for $k < 1$, this probability is actually $0$. For $k \approx 1$ I reckon this probability should be around $\frac{1}{n}$, and I think that if we replace $(\log(n))^k$ by $c \log(n)$ for some $c > 1$, then something close to $\frac{1}{n}$ should still hold, maybe $\frac{1}{n^{1 + o(1)}}$. In any case, I have no clue as to what happens for $k > 1$, so I would love to know.