  For a real number $k$, define $\pi_k(x) = \sum_{p\le x} p^k$. When $k>0$, we
prove that $$ \pi_k(x) - \pi(x^{k+1}) =
\Omega_{\pm}\left(\frac{x^{\frac12+k}}{\log x} \log\log\log x\right) $$ as
$x\to\infty$, and we prove a similar result when $-1<k<0$. This strengthens a
result in a paper by J. Gerard and the author and it corrects a flaw in a proof
in that paper. We also quantify the observation from that paper that $\pi_k(x)
- \pi(x^{k+1})$ is usually negative when $k>0$ and usually positive when
$-1<k<0$.
