由PSLQ算法求取数学常数的近似表达式：一种简单方法及实例研究
Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study
摘要：
在本文中，我提出了一个简单的PSLQ代码，用于在一定的精度范围内寻找数学常数的最佳有理系数的零线性组合。作为一个例子，我探索近似表达式的Apéry常数$\,ζ{(3)} = \sum_{n\ge1}{\,1/n^3}$，无理数，没有确切的，有限的封闭形式的表达是已知的。为此，我选择了一个合适的搜索基础，由似乎与$\,ζ{(3)}$密切相关的数字组成，即$\,π$，$\,\ln{2}\,$，$\,\ln{(1+\sqrt{2}\,)}$和$G$（加泰罗尼亚常数）。考虑到合适的搜索基础，我找到了一个精确到小数点后21位的$\,ζ{(3)}\,$简单表达式，比之前最好的表达式精确三倍。由于这里提供的简短的\emph{Maple} $^\mathrm{TM}$代码可以很容易地用于研究其他常数，我决定提供它以鼓励读者进行自己的计算实验，以及在数值分析，数论或线性代数的项目中采用它。
英文摘要：
In this note, I present a simple PSLQ code for finding null linear combinations, with the best rational coefficients, of mathematical constants, within some prescribed precision. As an example, I explore approximate expressions for the Apéry's constant $\,ζ{(3)} = \sum_{n\ge1}{\,1/n^3}$, an irrational number to which no exact, finite closed-form expression is known. % For this, I choose a suitable search basis composed by numbers which seem to be closely related to $\,ζ{(3)}$, namely $\,π$, $\,\ln{2}\,$, $\,\ln{(1+\sqrt{2}\,)}$, and $G$ (the Catalan's constant). On taking into account a suitable search basis, I have found a simple expression for $\,ζ{(3)}\,$ accurate to 21 decimal places, which is triply more accurate than the best previous one. As the short \emph{Maple}$^\mathrm{TM}$ code presented here can be easily adapted to study other constants, I decided to supply it to encourage the readers to conduct their own computational experiments, as well as to adopt it in projects of numerical analysis, number theory, or linear algebra.
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