Nesterenko-like rational function, useful to prove the Apéry’s theorem

Anier Soria Lorente
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 2, Pages 79—91
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Anier Soria Lorente
Department of Basic Sciences
Granma University, Cuba

Abstract

In this paper, a brief introduction to the Apéry’s result and to the so called phenomenon of Apéry’s is given. Here, a modification of the Nesterenko’s rational function, from which new Diophantine approximations to ζ(3) are deduced, is presented. Moreover, as a consequence we deduce the corresponding Apéry-Like recurrence relation as well as a new continued fraction expansion and a new series expansion for ζ(3).

Keywords

• Riemann zeta function
• Apéry’s approximants
• Recurrence relation
• Continued fraction expansion
• Irrationality

AMS Classification

• Primary: 11B37, 30B70, 14G10, 11J72, 11M06
• Secondary: 37B20, 11A55, 11J70, 11Y55, 11Y65

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