Erdős Problem 420

### What is the conjecture

https://www.erdosproblems.com/420

If $\tau(n)$ counts the number of divisors of $n$ then let
$$F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.$$
Is it true that
$$\lim_{n\to \infty}F((\log n)^C,n)=\infty$$
for large $C$? Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$? More generally, if $f(n)\leq \log n$ is a monotonic function then is $F(f,n)$ everywhere dense?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else