A new proof of the irrationality of \(\zeta (2)\) and \(\zeta (3)\) using Padé approximants
Back in the late seventies, Apéry's astonishing proof of the irrationality of \(\zeta (3)\) [see the reviewer, Math. Intell. 1, 195-203 (1979; Zbl 0409.10028)], raised cries of ``What is going on here?'' Such a question may of course have a variety of appropriate responses. Whatever, this paper provides a valuable such answer. Given a (formal) power series \(f= \sum^\infty_{h=0} c_h t^h\), in the author's notation a Padé approximant \([m, n]_f\) to \(f\) is a quotient of polynomials \(Q_m/ P_n\), with \(\deg Q_m =m\), \(\deg P_n= n\) and \(P_n (t) f(t)- Q_m (t)= O(t^{n+ m+1})\). One notices that \[ \zeta (s)= \sum^n_{k=1} k^{-s}+ {\textstyle {1\over {\Gamma (s)}}} \Psi_s (1/ n), \] where \(\Psi_s (x)= \int_0^\infty u^{s-1} {{e^{-u/x}} \over {e^u -1}} du\), and considers certain Padé approximants to \(\Psi_s\), with the intention of obtaining approximations to \(\zeta (s)\) by replacing \(\Psi_s (1/n)\). For example, in dealing with \(\zeta (3)\), the unknown \(\Psi_3 (1/n)\) is replaced by \([2n, 2n]_{\Psi_3} (x=1/n)\), yielding precisely the approximations arising in Apéry's proof. The core of the paper is the investigation of the Padé approximants in the cases \(s= 2, 3\), inter alia rediscovering the role that various hypergeometric polynomials can be said to play in the proof. The reviewer is frequently asked ``What about \(\zeta (5)\)?'' I could now try the author's answer that ``the `weight function' underlying the functions \(\Psi_s\) is no longer positive on its support when \(s\geq 4\), so Padé approximation is not appropriate for constructing approximations to \(\zeta (n)\) for \(n=5 \dots\)''.
scientific article; zbMATH DE number 908370
scientific article