HYPERGEOMETRIC CONSTRUCTIONS OF RATIONAL APPROXIMATIONS FOR (MULTIPLE) ZETA VALUES

This survey presents certain results concerning the diophantine nature of zeta values or multiple zeta values that I have obtained over the last few years, with or without coauthors. I did not try to cover all the known results concerning the diophantine theory of the Riemann zeta function and more informations are available in [12] for example. The first part is a presentation of irrationality results for the values of the Riemann zeta function, together with a description of the memoir [17] joint with Christian Krattenthaler on the “Denominators conjecture”. The second part describes some of the results in two papers with J. Cresson and S. Fischler [10, 11], both devoted to the construction of linear forms in multiple zeta values, which are generalisations of Riemann zeta function. I warmly thank K. Matsumoto and H. Tsumura, the organisers of the franco-japanese winter school in january 2008 at Miura seaside, for giving me the opportunity to publish this survey in the proceedings of that conference. In diophantine approximation, proofs of irrationality, linear independence, etc, usually rely on the construction of “auxiliary functions ” and this is also the case here. Indeed, the results presented in both parts of the paper are proved by means of explicit auxiliary functions, which turn out to be hypergeometric series in one or several variables. (The underlying aspect “Pade ́ approximants ” will not be developped.) Therefore, before going into the subject, it is useful to remind the reader of the definition of those series. These are power series defined by q+1Fq α0, α1,..., αq β1,..., βq; z k=