Sums of Powers of Primes II

For a real number $k$, define ${\pi }_k(x)={\sum }_{p\le x}{p}^k$. When $k>0$, we prove that $${\pi }_k(x)-\pi (x^{k+1})={\mathrm{\Omega }}_{\pm }(\frac{x^{\frac{1}{2}+k}}{\log x}\log \log \log x)$$ as $x\to \mathrm{\infty }$, 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 $-1<k<0$.