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

##### MSC:
 11J82 Measures of irrationality and of transcendence 41A21 Padé approximation 33C20 Generalized hypergeometric series, $${}_pF_q$$
##### Citations:
Zbl 0647.10002; Zbl 0679.10026
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