AMM Problem 12493 Solution

In this article, I provide the solution to AMM Problem 12493. This article is being published a month after the AMM deadline, which I had missed as I didn't see this earlier!
This problem is really interesting! It shows that a limit of the nth root of the number of divisors of 1!2!...n! is e.
It's an unexpected connection between analysis and Analytic number theory, that uses the prime number theorem and a few of its equivalences to complete the proof. If you know analytic number theory, or enjoy number theory in general, this proof is a great read!
Beware of the bash!

The proof can be found clicking here

I hope you enjoyed reading the article

Any feedback is greatly appreciated!