Let $\tau(k)$ be the number of divisors of the positive integer $k.$
How does $f(n)\stackrel{\triangle}{=}\prod_{k\leq n} \tau(k)$ or a reasonable function of it,such as $\log f(n)$ or $f(n)^{1/n}$ grow with increasing $n$? Any references, comments appreciated.
Partial Answer: I can see that from the arithmetic geometric mean inequality we have $$\prod_{k\leq n} \tau(k) \leq \left(\frac{\tau(1)+\cdots+\tau(n)}{n} \right)^n \approx (\log n+2\gamma-1+O(n^{-1/2}))^n \approx (\log n)^n \left(1+\frac{(2\gamma-1) n}{\log n}\right)$$ due to the well known asymptotic $$\tau(1)+\cdots+\tau(n)\approx n \log n +(2\gamma-1)n+O(\sqrt{n}).$$
The issue is, is there a tighter upper bound?
Here is a logarithmic plot of the upper bound approximation above and the actual product
Here is a plain plot of the ratio of the upper bound to the product, where the variability of the product with respect to the smooth upper bound is visible