On simultaneous Diophantine approximations to \(\zeta(2)\) and \(\zeta(3)\). (English) Zbl 1315.11062

Dauguet, Simon; Zudilin, Wadim

J. Number Theory 145, 362-387 (2014).

The authors construct simultaneous rational approximations to both \(\zeta(2)\) and \(\zeta(3)\) using hypergeometric tools. They prove that if \(\eta>0\) and \(\varepsilon>0\) are given, and \(m\) is sufficiently large with respect to \(\varepsilon\) and \(\eta\), then \(|a_0+ a_1\zeta(2)|> e^{-(s_0+ \eta)m}\) with \(s_0= 6.770732145\dots\), where \((a_0,a_1,a_3)\in \mathbb{Q}^3\setminus\{0\}\) satisfies certain strong divisibility conditions, and \(|a_0|,|a_1|,|a_2|\leq e^{-(\tau_0+\varepsilon)m}\) with \(\tau_0= 0.899668635\dots\). This result implies the irrationality of both \(\zeta(2)\) and \(\zeta(3)\), does not give however the expected linear independence of 1, \(\zeta(2)\) and \(\zeta(3)\).
Moreover, the authors further introduce a new notion of simultaneous Diophantine exponent, and give the basic properties of this new concept, and compare it with the classical irrationality exponent and S. Fischler’s \(\psi\)-exponent of irrationality [Indag. Math., New Ser. 20, No. 2, 201–215 (2009; Zbl 1198.11057)].

Reviewer: Yaochen Zhu (Beijing)

MSC:

11J82	Measures of irrationality and of transcendence
11J72	Irrationality; linear independence over a field
33C20	Generalized hypergeometric series, \({}_pF_q\)

Keywords:

irrationality; Diophantine approximation; irrationality exponent; hypergeometric series; hypergeometric integral; algorithm of creative telescoping

Citations:

Zbl 1198.11057

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