论文标题
在第一个Chebyshev功能上有效的De la Valle Poussin风格界限
Effective de la Valle Poussin style bounds on the first Chebyshev function
论文作者
论文摘要
1898年，查尔斯·迪拉·瓦莱·普森（Charles Jean de la Valle Poussin）作为他著名的素数定理证明的一部分，在形式的第一个chebyshev函数上产生了无效的界限：\ [|θ（x）-x | = \ Mathcal {o} \ left（x \ exp（-k \ sqrt {\ ln x}））\ right）。 \]此限制为$ x $足够大，$ x \ geq x_0 $和$ k $有些未指定的正常常数。据我所知，这种界限从未有效 - 我从未见过这种界限完全明确，并以$ x_0 $和$ k $给出精确的值。此处，使用过去50年中建立的许多有效结果，我将开发两个非常简单的这种类型的完全有效界限：\ [|θ（x）-x | <\; {x} \; \ exp \ left（ - {1 \ over4} \ sqrt {\ ln x} \ right）; \ qquad（x \ geq 2）。 \] \ [|θ（x）-x | <\; {x} \; \ exp \ left（ - {1 \ over3} \ sqrt {\ ln x} \ right）; \ qquad（x \ geq 3）。 \]可以很容易地开发出许多其他沿着这些线路的完全明确的界限。例如，可以在有效性范围内进行严格权衡：\ [|θ（x）-x | <\; {1 \ over 2} \; {x} \; \ exp \ left（ - {1 \ over4} \ sqrt {\ ln x} \ right）; \ qquad（x \ geq 29），\] \ [|θ（x）-x | <\; {1 \ over 2} \; {x} \; \ exp \ left（ - {1 \ over3} \ sqrt {\ ln x} \ right）; \ qquad（x \ geq 41）。 \ \]在事后看来，其中一些有效的界限本可以在大约50年前建立。
In 1898 Charles Jean de la Valle Poussin, as part of his famed proof of the prime number theorem, developed an ineffective bound on the first Chebyshev function of the form: \[ |θ(x)-x| = \mathcal{O}\left(x \exp(-K \sqrt{\ln x})\right). \] This bound holds for $x$ sufficiently large, $x\geq x_0$, and $K$ some unspecified positive constant. To the best of my knowledge this bound has never been made effective -- I have never yet seen this bound made fully explicit, with precise values being given for $x_0$ and $K$. Herein, using a number of effective results established over the past 50 years, I shall develop two very simple explicit fully effective bounds of this type: \[ |θ(x)-x| < \; {x} \;\exp\left( - {1\over4} \sqrt{\ln x}\right); \qquad (x\geq 2). \] \[ |θ(x)-x| < \; {x} \;\exp\left( - {1\over3} \sqrt{\ln x}\right); \qquad (x\geq 3). \] Many other fully explicit bounds along these lines can easily be developed. For instance one can trade off stringency against range of validity: \[ |θ(x)-x| < \; {1\over 2} \; {x} \;\exp\left( - {1\over4} \sqrt{\ln x}\right); \qquad (x\geq 29), \] \[ |θ(x)-x| < \; {1\over 2} \; {x} \;\exp\left( - {1\over3} \sqrt{\ln x}\right); \qquad (x\geq 41). \] With hindsight, some of these effective bounds could have been established almost 50 years ago.
