Sums of powers of primes.

Let $$k>-1$$ . The sum of the kth powers of the primes less than x is asymptotic to $$\pi (x^{k+1})$$ . We show that the sum is less than $$\pi (x^{k+1})$$ for arbitrarily large x, and the reverse inequality also holds for arbitrarily large x. When $$k>0$$ , there is a bias toward the first inequality, and we explain why this should be true and why the reverse bias holds when $$-1<k<0$$ . [ABSTRACT FROM AUTHOR]