The Summation of x(log(log(x))

Question

So I would like to know if it is possible to express this summation in terms of $n$:

$$\sum_{x=2}^n x\log(\log(x))$$

For example the summation below is equivalent to $\frac12 n (n+1)$ in terms of $n$

$$\sum_{x=0}^nx=\frac12n(n+1)$$

Answer 1

You can use some logarithm identities to achieve a little progress, but I would be surprised if there is anything better. $$\sum\limits_{x=2}^nx\log(\log(x))=\sum\limits_{x=2}^n\log(\log^x(x))=\log\left(\prod\limits_{x=2}^n\log^x(x)\right)$$