Sums of Powers of Primes II

TL;DR

This paper investigates the oscillatory behavior of sums of prime powers, establishing lower bounds for their deviations from prime counting functions and clarifying their typical signs depending on the exponent k.

Contribution

It proves new lower bounds for the difference between sums of prime powers and prime counts, correcting previous proofs and extending understanding of their sign behavior.

Findings

Established Omega results for prime power sums for k>0 and -1<k<0.

Corrected a flaw in a previous proof by Gerard and the author.

Quantified the typical sign of the difference based on the value of k.

Abstract

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 $-1<k<0$.