I am interested in an upper bound for $$ \sum_{\substack{d|N\\ d>A}}\frac{1}{d^3},$$ in particular, I can get the above to be $$\sum_{\substack{d|N\\ d>A}}\frac{1}{d^3}\ll \frac{\text{exp}\left(C \frac{\log(N)}{\log\log(N)}\right)}{A^3}$$ for some positive constant $C$. However I would like to do better. I think that the upper bound should be around $$ \frac{\log^C(N)}{A^3}$$ for some other positive constant $C.$
We also have that $A<N^{1/2}.$ References are welcome.