Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study

TL;DR

This paper introduces a simple PSLQ algorithm implementation to find highly accurate approximate expressions for mathematical constants, demonstrated on Apéry's constant, with potential applications in numerical analysis and number theory.

Contribution

The paper presents an accessible PSLQ code for deriving precise approximate expressions of constants, exemplified by a highly accurate approximation of Apéry's constant using a specific basis.

Findings

Achieved a 21-decimal-place accurate approximation of ζ(3).

Provided a flexible Maple code for exploring other constants.

Demonstrated the effectiveness of the approach with a case study.

Abstract

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 Ap\'{e}ry's constant $\,\zeta{(3)} = \sum_{n\ge1}{\,1/n^3}$, 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 $\,\zeta{(3)}$, namely $\,\pi$, $\,\ln{2}\,$, $\,\ln{(1+\sqrt{2}\,)}$, and $G$ (the Catalan's constant). On taking into account a suitable search basis, I have found a simple expression for $\,\zeta{(3)}\,$ 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.