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

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)].

### MSC:

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

Zbl 1198.11057
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### References:

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