Asymptotic behaviour of log log sum

Question

I am interested in the asymptotics of $$F(m) := \prod_{j=1}^m \log(j+1) = \exp\left(\sum_{j=1}^m \log \log(j+1) \right)$$

Is there anything known? If not I figure I will need some good bounds on the $\log\log$-function? Thanks!

Answer 1

This could be helpful for developing the asymptotic behaviour of your sum:

Define $f(x)=\log \log (x)$, then using this link you may write $$\sum_{j=2}^{m-1}\log \log (j)=\log \log2+\int_2^{m-1}f(x)dx+B_1(f(m-1)-f(2))+\sum_{k=1}^p\Big(f^{(2k-1}(m-1))-f^{(2k-1)}(2)\Big)\frac{B_{2k}}{(2k)!}+R$$

$B$'s are Bernoulli numbers, see the link above.