This was asked in mathstackexchange (see here) but was not satisfactorily answered beyond my basic observations.

Let $\tau(k)$ be the number of divisors of the positive integer $k$. How does $f(n)=\prod_{k\leq n} \tau(k)$ or a reasonable function of it, such as $\log f(n)$ grow with increasing $n$?

Gerry Myerson commented that it's tabulated at OEIS but with no information on growth rate.

I noted that the upper bound (using the arithmetic geometric mean inequality and the sum of divisors of integers up to $n$) below holds $$f(n)\leq (\log n)^n \left(1+\frac{2 \gamma -1}{n}\right).$$

This is a plot of $\log U$ and $\log f(n)$ where $U$ is the upperbound in (1).

This is a plot $U/f(n)$ where $U$ is the upperbound in (1).