On the measure of irrationality of the number \(\pi \) (Q650318)

The author proves that for all \( p,q\in\mathbb N\), \(q\geq q_0\), the inequality \(|\pi- p/q|> q^{-\nu}\) (where \(\nu=7.606308\dots\)) holds, and so gives a new estimate of the measure of irrationality of the number \(\pi\). This result improves the previous record \(\nu= 8.016045\dots\) by \textit{M. Hata} [Acta Arith. 63, No. 4, 335--349 (1993; Zbl 0776.11033)]. In order to prove this result, the author considers the integral \[ J= (1/i) \int^{4+2i}_{4-2i} R(x)\,dx, \] where \[ R(x)= \sum^{5n-2}_{i=0} b_i x^i+ \sum^{5n}_{i=0} (a_i x^{-i-1}+ a_i(10- x)^{-i-1}), \] and uses the saddle-point method.