TITLE: how to prove this extended prime number theorem?
QUESTION [12 upvotes]: A Generalized Prime Number Theorem?
Conjecture
Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest odd prime $\le n$, 
$$
3^k + 5^k + 7^k + 11^k + ... + p^k \sim \frac{n^{k+1}}{(k+1) ( \log(n) - \log(k) ) }
$$
I wonder if you guys have seen it before ?
How to prove it ?
Any useful references for $k > 1$ ?

REPLY [2 votes]: A slightly weaker form is the direct application of:

Theorem: Let $p_k$ be the $k$-th prime and let $f$ be a continuous function Riemann
  integrable in $(0,1)$ then, 
$$ \lim_{m \to \infty}\frac{1}{m}\sum_{r = 1}^{m}f\Big(\frac{p_r}{p_m}\Big) = \int_{0}^{1}f(x)dx. $$

For $f(x) = x^k$ we get the more elegant form
$$
2^k + 3^k + 5^k + \ldots + p_m^k \sim \frac{mp_m^k}{k+1} \sim \frac{m^{k+1} \log^k m}{m+1}
$$
and then express it in terms $n$ using $m \log m \sim n$.