Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding it, but does the function above seem to ring any bells. Any information would be helpful.

I conjecture that for all complex $z$ with $Re(z)\ge-1$, $$\sum\limits_{p\le n}p^z\approx\int\limits_{2}^{n}\frac{x^z}{\ln(x)}dx$$ Perhaps this makes things more interesting.