# On the sum of prime powers

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.

• Why are you interested? Are you looking for something special? – draks ... Apr 1 '14 at 20:59
• I was investigating the prime zeta function and ended up making the above conjecture which fitted very well with computation. So I just wanted to know who else had been researching these sums and well, no sea of research have I met, but a veritable jungle indeed! – Elie Bergman Apr 1 '14 at 21:25

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...
• @ElieBergman I can try: $d\pi(t)$ is a measure. Let me cite Ilya's comment "the integral is well-defined in the Lebesgue-Stieltjes sense." – draks ... Apr 1 '14 at 21:30