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.


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