The irrationality measure of \(\pi\) is at most 7.103205334137…. (English) Zbl 1456.11129

Zeilberger, Doron; Zudilin, Wadim

Mosc. J. Comb. Number Theory 9, No. 4, 407-419 (2020).

The main result of this paper is that the irrationality measure exponent of the number \(\pi\) is less than \(7.103205334138\). The proof uses complex analysis, is based on clever calculating of special integral and is in the spirit of Salikov.

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J82	Measures of irrationality and of transcendence
11Y60	Evaluation of number-theoretic constants
33F10	Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
33C60	Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)

Keywords:

\( \pi \); irrationality measure exponent; experimental mathematics; Almkvist-Zeilberger algorithm