Cited here, Pierre Dusart established the following lower bound for $p_n$:
$$p_n > n(\ln n + \ln \ln n - 1)$$
Using a spreadsheet and plugging in different values of $n$, I noticed that for an integer $a > 1$:
$$\frac{an(\ln (an) + \ln \ln (an) - 1)}{n(\ln n + \ln \ln n - 1)} > a$$
I am interested in figuring out if this is always true as a way of better understanding Dusart's estimate.
Would the right approach be to show that it is true for some $n$ and then to show the the derivative of:
$$\frac{ax(\ln (ax) + \ln \ln (ax) - 1)}{x(\ln x + \ln \ln x - 1)}$$
is positive? Is there a more appropriate way to verify this?
The part that confuses me is how to check this against any positive integer $a$.