Recent Diophantine results on zeta values: a survey

After the proof by R. Apéry of the irrationality of ζ(3) in 1976, a number of articles have been devoted to the study of Diophantine properties of values of the Riemann zeta function at positive integers. A survey has been written by S. Fischler for the Bourbaki Seminar in November 2002 [6]. Here, we review more recent results, including contributions by P. Bundschuh, S. Bruiltet, C. Elsner, S. Fischler, S. Gun, M. Hata, C. Krattenthaler, R. Marcovecchio, R. Murty, Yu.V. Nesterenko, P. Philippon, P. Rath, G. Rhin, T. Rivoal, S. Shimomura, I. Shiokawa, C. Viola, W. Zudilin. We plan also to say a few words on the analog of this theory in finite characteristic, with works of G. Anderson, W.D. Brownawell, M. Pappanikolas, D. Thakur, Chieh-Yu Chang, Jing Yu. 1 Special values of the Riemann zeta function Several zeta functions exist, including Riemann zeta function, Multizeta functions, Weierstraß zeta function, those of Fibonacci, Hurwitz, Carlitz, Dedekind, Hasse-Weil, Lerch, Selberg, Witten, Milnor and the zeta functions of dynamical systems. . . 1.1 The Riemann zeta function We first review the Riemann zeta function, which was previously introduced by L. Euler: ζ(s) = ∑ n≥1 1 ns for s ∈ R, s ≥ 2. He showed the Euler product : ζ(s) = ∏ p 1 1− p−s · ∗Notes written by N. Hirata from the text of the slides of the lecture given at the Conference on Analytic number theory and related topics, RIMS, Kyoto, Japan, organized by H. Tsumura. The author wishes to express his deep gratitude to Noriko Hirata and Hirofumi Tsumura. This text is available on the web site of the author at the address http://www.math.jussieu.fr/∼miw/articles/pdf/ZetaValuesRIMS2009.pdf