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).
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