论文标题
Primes II的权力总和
Sums of Powers of Primes II
论文作者
论文摘要
对于实际数字$ k $，定义$π_k（x）= \ sum_ {p \ le x} p^k $。当$ k> 0 $时，我们证明$$π_k（x） - π（x^{k+1}）=ω_ {\ pm} \ left（\ frac {x^{x^{\ frac12+k}}}} {\ log x} {\ log x}} $ -1 <k <0 $。这加强了J. Gerard和作者的论文中的结果，并在该论文中的证据中纠正了缺陷。我们还从该论文中量化了$π_k（x）-π（x^{k+1}）$通常为负时，当$ k> 0 $时，通常为$ -1 <k <0 $时，通常为负。
For a real number $k$, define $π_k(x) = \sum_{p\le x} p^k$. When $k>0$, we prove that $$ π_k(x) - π(x^{k+1}) = Ω_{\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 $π_k(x) - π(x^{k+1})$ is usually negative when $k>0$ and usually positive when $-1<k<0$.
