Ramanujan-type formulae and irrationality measures of some multiples of \(\pi\). (English. Russian original) Zbl 1114.11064

Zudilin, V. V.

Sb. Math. 196, No. 7, 983-998 (2005); translation from Mat. Sb. 196, No. 7, 51-66 (2005).

Basing on an explicit construction of simultaneous Padé approximations for generalized hypergeometric series and using the program proposed by D. V. Chudnovsky and G. V. Chudnovsky [Ramanujan revisited, Academic Press, Boston, MA, 375–472 (1988; Zbl 0647.10002); Theta functions, Part 2. Amer. Math. Soc., Providence, RI, 167–232 (1989; Zbl 0679.10026)], the author gives estimates of irrationality measures of the numbers \(\pi\sqrt d\), \(d\in\{1,2,3,10005\}\). Denoting the irrationality exponent of a real irrational number \(\alpha\) by \(\mu (\alpha)\), the main results are as follows: \[ \mu(\pi)\leq 57.53011083 \dots, \mu(\pi\sqrt 2)\leq 13.93477619\dots, \]
\[ \mu(\pi\sqrt 3)\leq 44.12528464\dots,\mu(\pi \sqrt{10005})\leq 10.02136339\dots. \] The last one among them refines the already known estimate (i.e., \(\mu(\pi\sqrt d)\leq 10.88248501\dots\) for any positive integer \(d)\). Moreover, it is worthy to note that the methods used in the present paper can find further number-theoretic applications.

Reviewer: Zhu Yaochen (Beijing)

MSC:

11J82	Measures of irrationality and of transcendence
41A21	Padé approximation
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

simultaneous Padé approximations; generalized hypergeometric series

Citations:

Zbl 0647.10002; Zbl 0679.10026