How to find an asymptotic formula for $f(n)=\sum_{k=1620}^{n}(\log\log\log k)^{2}$?

Question

How to find an asymptotic formula for function given below.

$$f(n)=\sum_{k=1620}^{n}(\log\log\log k)^{2}$$

Answer 1

It is asymptotic to $n (\log\log\log n)^2$. This is clearly an upper bound. To see that it is a lower bound, compute the sum of $\sum_{\sqrt n}^n$. This is clearly bounded below by $(n-\sqrt n) (\log\log\log \sqrt n)^2$, and for large $n$ the ratio of this quantity and $n (\log\log\log n)^2$ converges to $1$.