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.
Hmm, I called it the truncated Prime zeta function. In a general way you can write any function that sums over primes like this $$ \sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t)) =f(t)\pi(t)\biggr|_{2}^{x}+\int_{2}^{x}f'(t)\pi(t)dt. $$ see here. Put in your favorite approximation for $\pi(n)$, like $\frac n{\log n}$, and $f(t)=t^z$ you get: $$ \sum_{p\le x} p^z \approx \frac {t^{z+1}}{\log t}\Biggr|^x_2 +\int_2^x z\frac {t^{z}}{\log t} dt $$ close to what you conjecture...
