<< The last time I looked, essentially nothing was known about the odd zeta's. In fact, zeta(3) was shown irrational in 1978 by Roger Apéry. In 2001, Tanguy Rivoal showed that there are infinitely many odd (positive) integers at which zeta is irrational, including at least one value j in the range 5 <= j <= 21 (refined the same year by Zudilin to 5 <= j <= 11), at which zeta(j) is irrational. See < http://algo.inria.fr/seminars/sem01-02/rivoal.ps > for proofs of Rivoal's results, and citations for the others. (A classical result is that zeta(N) is rational for N <= 0, with zeta(2N) = 0 for 2N < 0.) --Dan