On the measure of irrationality of the number \(\pi \). (English. Russian original) Zbl 1257.11072
Math. Notes 88, No. 4, 563-573 (2010); translation from Mat. Zametki 88, No. 4, 583-593 (2010).
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 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.
Reviewer: Zhu Yaochen (Beijing)
MSC:
| 11J82 | Measures of irrationality and of transcendence |
| 11J72 | Irrationality; linear independence over a field |
Keywords:
measure of irrationality; approximation by rational fractions; partial-fraction expansion; Leibniz formula; linear formCitations:
Zbl 0776.11033References:
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