Arithmetic of linear forms involving odd zeta values par WADIM ZUDILIN
نویسنده
چکیده
RÉSUMÉ. Une construction hypergéométrique générale de formes linéaires de valeurs de la fonction zéta aux entiers impairs est présentée. Cette construction permet de retrouver les records de Rhin et Violla pour les mesures d’irrationnalité de 03B6(2) et 03B6(3), ainsi que d’expliquer les résultats récents de Rivoal sur l’infinité des valeurs irrationnelles de la fonction zéta aux entiers impairs et de prouver qu’au moins un des quatre nombres 03B6(5), 03B6(7), 03B6(9) et 03B6(11) est irrationnel.
منابع مشابه
Arithmetic of Linear Forms Involving Odd Zeta Values
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 ζ(2) and ζ(3), 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 ζ(5), ζ(7), ζ(9), and ζ(11) i...
متن کامل2 1 Ju n 20 02 Arithmetic of linear forms involving odd zeta values ∗
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 ζ(2) and ζ(3), as well as to explain Rivoal’s recent result (math.NT/0008051) on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers ζ(5), ζ(7),...
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It is explained how the classical concept of well-poised hypergeometric series and integrals becomes crucial in studing 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 ζ(4) = π/90 yielding a conditional upper bound for the irrationality measure of ζ(4); (2) a second-order Apéry-like recursion...
متن کاملArithmetics of Linear Forms Involving Odd Zeta Values
The story exposed in this paper starts in 1978, when R. Apéry [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apéry’s discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apéry’s proof allows ...
متن کاملJu n 20 02 Arithmetic of linear forms involving odd zeta values ∗
The story of this work is very dramatic for me. The paper is rejected twice by good mathematical journals, although no negative reports were obtained by editorial boards. The general advice of referees and other mathematicians is to cut this article and to write a shorter paper stressing reader’s attention on Sections 7, 8 and Theorem 3 (“at least one of the four numbers ζ(5), ζ(7), ζ(9), and ζ...
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تاریخ انتشار 2006