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Hypergeometry and Riemann’s zeta function. (Hypergéométrie et function zêta de Riemann.) (English) Zbl 1113.11039
Mem. Am. Math. Soc. 875, 87 p. (2007).
This interesting book deals with the proofs when the several rational numbers are integers. These numbers concern the common denominators of the coefficients of the certain linear forms in zeta values. As a consequence the authors demonstrate that at least one of the eight numbers \(\zeta(5),\zeta(7),\dots, \zeta(19)\) is irrational or there exists an integer \(j\) between \(5\) and \(165\) such that \(1\), \(\zeta(3)\) and \(\zeta(j)\) are linearly independent over \(\mathbb{Q}\). Another nice consequence is an approximation of \(\zeta(4)\) by rationals. The proofs are in the spirit of Apéry using algebraic and combinatorial methods and they are based on a special hypergeometric identity. The authors believe that maybe in this way it will be possible to prove e.g. that at least one number among \(\zeta(5)\), \(\zeta(7)\) and \(\zeta(9)\) is irrational.

MSC:
11J72 Irrationality; linear independence over a field
11J82 Measures of irrationality and of transcendence
33C20 Generalized hypergeometric series, \({}_pF_q\)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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