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

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 $1$.
