Sums of powers of primes II

For a real number k, define pi(k)(x)=& sum;(p <= x)p(k). When k > 0, we prove that pi(k)(x)-pi(x(k+1))=Omega +/-(x(1/2+k /)logx log log logx) asx ->infinity, 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 pi(k)(x)-pi(x(k+1))is usually negative when k > 0 and usually positive when-1 < k < 0.