Similarities in irrationality proofs for \(\pi\), \(\ln 2\), \(\zeta(2)\), and \(\zeta(3)\). (English) Zbl 0986.11045

This paper develops a strategy suggested by work of the Borweins [J. M. Borwein and P. B. Borwein, Pi and the AGM. A study in analytic number theory and computational complexity, Wiley, New York (1987; Zbl 0611.10001)] for deducing the irrationality of a given number \(\xi\). Find a function \(f\) and a sequence of rational numbers \(R_j\) and \(S_j\) such that \(\int_0^1 x^jf(x) dx= R_j+\xi S_j\). If \(\xi\) is rational, then any finite linear combination of \(R_j+ \xi S_j\) with integer multipliers is also rational. Use the coefficients \(p_{nj}\) of the \(n\)th Legendre polynomial \(P_n(x)\) as multipliers, and sum on \(j\) to get \(A_n= B_n\int_0^1 P_n(x)f(x) dx\) where \(A_n\) and \(B_n\) are integers. For each number in the title one can choose a suitable \(f\) and obtain an inequality of the form \(1\leq|A_n|\leq C_n\), where \(C_n\to 0\) as \(n\to\infty\). This is a contradiction.


11J72 Irrationality; linear independence over a field


Full Text: DOI