Part VIII Hypergeometric constructions of rational approximations for (multiple) zeta values

摘要

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 will not try to cover all the known results concerning the diophantine theory of the Riemann zeta function and more information is 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 present lecture notes.