论文标题
质数定理中错误术语的更清晰界限
Sharper bounds for the error term in the Prime Number Theorem
论文作者
论文摘要
我们提供了非常有效的方法，可以将prime计数函数上的渐近数字和显式数字界转换为$θ（x）$和$π（x）$的相同类型的边界。这遵循了我们以前在\ cite {fks}中$ψ（x）$的工作，并证明了$ | π（x） - \ mathrm {li}（x）| \ leq 9.2211 \，x \ sqrt {\ log（x）} \ exp \ big big（-0.8476 \ sqrt {\ log（x）} \ big）$ for $ x \ ge 2 $。此外，我们能够在非常大的间隔（全部$ x $ to $ \ exp（1.8 \ cdot10^9）$上获得$ x $的最佳数字界。
We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function $ψ(x)$ to bounds of the same type on both $θ(x)$ and $π(x)$. This follows up our previous work on $ψ(x)$ in \cite{FKS}, and prove that $ | π(x) - \mathrm{Li}(x) | \leq 9.2211\, x\sqrt{\log(x)} \exp \big( -0.8476 \sqrt{\log(x)} \big) $ for all $x\ge 2$. Additionally, we are able to obtain the best numeric bounds for $x$ on a very large interval (all $x$ up to $\exp(1.8\cdot10^9)$).
