I have a question about the estimate of the divisor function. Let $$ d(n)=\sum_{d|n}1.$$
I proved that $$\sum_{n<x}d(n) \ll x \log x \\ \sum_{n<x}\frac{d(n)}{n} \ll (\log x)^2.$$
My question is whether we can get the estimate $$d(n) \ll_{\varepsilon} n^{\varepsilon} $$ for any $\varepsilon >0$ from above two inequality.
If you have an idea, please tell me about my question.