From miw@mathp6.jussieu.frWed Oct 2 16:14:51 1996 Date: Wed, 2 Oct 1996 09:01:23 -0400 From: Michel Waldschmidt To: NMBRTHRY@listserv.nodak.edu Subject: sum n^{-5} >From: David Epstein >Is it still unknown whether \sum n^{-5} is a rational number? Yes, it is still an open problem. One expects that the answer is: \sum n^{-5} is an irrational number. Other similar open problems are: is Euler constant \gamma an irrational number? is e^\gamma an irrational number? is e+\pi an irrational number? is \Gamma(1/5) an irrational number? On the other hand, there is some recent progress on this matter. For instance it was proved in 96 (by Yu. V. Nesterenko) that \pi+e^\pi is irrational (in fact, he proved that \pi, e^\pi and \Gamma(1/4) are algebraically independent - but before that it was not known that \pi+e^\pi is irrational). Nesterenko's result also includes the transcendence of \sum 2^{-n^2} (whose irrationality was noticed by Liouville in 1844 - in fact irrationality is obvious, just look at the 2-adic expansion; but Liouville gave another argument). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Michel WALDSCHMIDT Institut de Mathematiques de Jussieu % % Problemes Diophantiens case 247 F-75252 PARIS Cedex 05 % % Tel: 44 27 53 36 Fax: 48 44 - Secretariat: 53 44 Fax: 73 21 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%