Sums of powers of primes II

For a real number k , define π k ( x ) = ∑ p ≤ x p k . When k > 0 , we prove that π k ( x ) - π ( x k + 1 ) = Ω ± x 1 2 + k log x log log log x as x → ∞ , and we prove a similar result when - 1 < k < 0 . This strengthens a result in a paper by Gerard and the author and it corrects a flaw in a proof in that paper. We also quantify the observation from that paper that π k ( x ) - π ( x k + 1 ) is usually negative when k > 0 and usually positive when - 1 < k < 0 .