Products of the iteration of the divisor function

Let $d(n)$ denote the number of divisors of $n$. Erd{\H{o}}s, in 1968, proved the remarkable result whereby there exists a positive constant $ d_{2} < \infty$ such that $$ \lim_{x \to \infty} \frac{\sum_{n \leq x} d(d(n))}{x \log \log x} = d_{2}, $$ and this was improved by Rieger in 1972. If we compare the above result to Ramanujan's formula for $ \prod_{n=1}^{x} d(n)$ and its relation to the Dirichlet formula for $\sum_{n = 1}^{x} d(n)$, this raises questions as to the behaviour of products of the form $\prod_{n=1}^{x} d(d(n))$. By the AM-GM inequality, Erd{\H{o}}s' result gives us that $$ \frac{\left( \prod_{n=1}^{x} d(d(n)) \right)^{\frac{1}{x}}}{\log \log x} $$ is bounded above by a positive, finite constant, but this does not give that the limit of the above expression exists as a positive constant, and known bounds for expressions such as $d(n)$ and $\log d(d(n))$ cannot be used in any direct way. Moreover, as we discuss, it seems that Erd{\H{o}}s' and Rieger's summation techniques cannot be altered so as to be applicable to expressions such as $\sum_{n \leq x} \log d(d(n))$ or $\prod_{n \leq x} d(d(n))$. We prove that: For all $\epsilon > 0$, the bounds $$ \log 2 - \epsilon \leq \frac{\sum_{n \leq x} \log d(d(n)) }{x \log \log \log x} \leq 1 + \epsilon $$ hold for all for sufficiently large $x$, yielding bounds for the product obtained by replacing $d(n)$ with $d(d(n))$ in Ramanujan's product.

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