I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$
I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$
For my application, I need something like $$\sum_{\substack{d | n\\ d \geq N}} 1 \leq \frac{o(n^{\epsilon})}{\log N} \quad \forall \epsilon > 0. $$
A reference where the bound can be found or a simple proof would be appreciated.
Thanks.
EDIT: Johan Andersson at mathoverflow has pointed out that the third display follows from the second. (Thanks.) I am still interested to learn what the best known bound is.