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