Asymptotics for this sum...

Question

$$\sum_{\substack{p \leq x \\ p=\left \lfloor n^{d} \right \rfloor}}\frac{1}{p}$$ with $0<d<1$; $d\in \mathbb{R}$; $n,p \in \mathbb{N}$

Can you give an asymptotics for this sum? Thank you.

Answer 1

If $d<1$, $n^d$ is more dense than $n$ and your sum is $H_{\lfloor x\rfloor}$ of which the asymptotics is well known.