## Well-poised hypergeometric series for Diophantine problems of zeta values.(English)Zbl 1156.11326

Summary: It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studying arithmetic properties of the values of Riemann’s zeta function. By these well-poised means we obtain: (1) a permutation group for linear forms in 1 and $$\zeta(4)=\pi^4 /90$$ yielding a conditional upper bound for the irrationality measure of $$\zeta(4)$$; (2) a second-order Apéry-like recursion for $$\zeta(4)$$ and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family of certain Euler-type multiple integrals that generalize so-called Beukers’ integrals for $$\zeta(2)$$ and $$\zeta(3)$$.
Note: There is an obvious misprint in the title: replace service by series.

### MSC:

 11J72 Irrationality; linear independence over a field 11J81 Transcendence (general theory) 33C20 Generalized hypergeometric series, $${}_pF_q$$

### Keywords:

well-poised hypergeometric series
Full Text:

### References:

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