Is there any way to find the following limit

$$R(n,m)=\lim_{N\to\infty}\frac{H_{nN,m}}{H_{N,m}}$$

which involves harmonic numbers (generalized if $m\neq 1$)

$$H_{N,m}=\sum_{k=1}^N k^{-m}\qquad ?$$

I am more specifically looking for a convenient way to compute it numerically for $m<1$ (if it converges to something else than 1 of course).

For instance we can guess $R(n,1/2)=\sqrt{n}$ using Mathematica.