论文标题
Chebyshev函数$ψ（x）$的更清晰的边界
Sharper bounds for the Chebyshev function $ψ(x)$
论文作者
论文摘要
我们改善了Prime Counting函数$ψ（x）$中错误项的无条件明确界限。特别是，我们证明，对于所有$ x> 2 $，我们有\ [\ left | ψ（x）-x \ right | <9.22106 \，x \，（\ log x）^{3/2} \ exp（-0.8476836 \ sqrt {\ log x}），\]，对于所有$ \ log x \ ge 3 \， ψ（x）-x \ right | <4.47 \ cdot 10^{ - 15} x。 \]这与Platt \＆Trudgian（2021）的结果相比，他获得了$ 4.51 \ cdot 10^{ - 13} x $。我们的方法代表了Platt和Trudgian应用的Pintz观念的重要改进。改进是通过将零分配给其他区域，仔细估算所有随之而来的术语以及大量使用计算方法来获得的改进。有关$π（x）$的结果将出现在后续工作中。
We improve the unconditional explicit bounds for the error term in the prime counting function $ψ(x)$. In particular, we prove that, for all $x>2$, we have \[ \left| ψ(x)-x \right| < 9.22106 \, x \, (\log x)^{3/2} \exp(-0.8476836\sqrt{\log x}), \] and that, for all $\log x \ge 3\,000$, \[ \left| ψ(x)-x \right| < 4.47\cdot 10^{-15} x. \] This compares to results of Platt \& Trudgian (2021) who obtained $4.51\cdot 10^{-13} x $. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning $π(x)$ will appear in a follow up work.
