Sums of Powers of Primes II

Lawrence C. Washington

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arXiv · 2209.12845

Sums of Powers of Primes II

Lawrence C. Washington

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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 $-1 0$ and usually positive when $-1<k<0$.

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Lawrence C. Washington. 2022-09-22. Sums of Powers of Primes II. https://arxiv.org/abs/2209.12845

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