I am interested in asymptotics for the logarithmic moment (is that the right way to express this) of the sum of squares function $r_2(n)$ (https://en.wikipedia.org/wiki/Sum_of_squares_function), that is: \begin{equation} \sum_{n=1}^{N} r_2(n) \log(n) \end{equation} for large $N$.
I have seen similar questions relating to integer moments \begin{equation} \sum_{n=1}^{N} r_2(n) n^m, \end{equation} how ever I am unsure how these extend in the $\log$ case.
