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 and yielding a conditional upper bound for the irrationality measure of ; (2) a second-order Apéry-like recursion for and some low-order recursions for linear forms in odd zeta values; (3) a rich permutation group for a family...
We prove that, for all integers exceeding some effectively computable number , the distance from to the nearest integer is greater than .
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of and , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers , , , and is irrational.
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