Let $d(n!)$ be the number of divisors of $n!$. I observed that graph of $\log(d(n!)$ is similar to that of $\pi(x)$, the number of prime not exceeding $x$ except for a scaling factor. In other words, the following limit must exist: $$ \lim_{n \to \infty}\frac{\log(d(n!))}{\pi(n)} \approx 1.26 $$
Question: What is known about the number of divisors of $n!$? Can it be proved or disproved?
A short argument of Erdős et al. (see $\S$2 of the cited article) shows that
$$\log d(n!) \sim \frac{c_0 \log (n!)}{\log^2 \log (n!)} \left[1 + O\left(\frac{\log\log\log (n!)}{\log\log (n!)}\right)\right],$$ where $$\color{#df0000}{c_0 = \int_1^\infty \frac{\log (\lfloor t \rfloor + 1) \,dt}{t^2} = \sum_{k = 2}^\infty \frac{\log k}{k (k - 1)} = 1.25775\!\ldots }.$$ Applying Stirling's Approximation then gives that the leading term of the above series approximation is $$\frac{c_0 n}{\log n} + O\left(\frac{n \log \log n}{(\log n)^2}\right) .$$ On the other hand, the Prime Number Theorem says that $$\pi(n) \sim \frac{n}{\log n} + o \left(\frac{n}{\log n}\right),$$ so $$\color{#df0000}{\boxed{\lim_{n \to \infty} \frac{\log d(n!)}{\pi(n)} = c_0}} .$$
Remark In fact, we can compute higher asymptotics of the ratio using the formulae of Erdős et al. (and a better bound on the remainder of the above approximation to $\pi(n)$), e.g., $$\frac{\log d(n!)}{\pi(n)} \sim c_0 + \frac{c_1}{\log n} + O\left(\frac{1}{\log^2 n}\right), $$ where $$c_1 := \int_1^\infty \frac{\log(\lfloor t \rfloor + 1) \log t \,dt}{t^2} = 2.11412\!\ldots .$$