  We show that the Mobius function mu(n) is strongly asymptotically orthogonal
to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected
nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a
nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a
Lipschitz function. More precisely, we show that the inner product of mu(n)
with F(g(n)L) over {1,...,N} is bounded by 1/log^A N, for all A > 0. In
particular, this implies the Mobius and Nilsequence conjecture MN(s) from our
earlier paper "Linear equations in primes" for every positive integer s. This
is one of two major ingredients in our programme, outlined in that paper, to
establish a large number of cases of the generalised Hardy-Littlewood
conjecture, which predicts how often a collection \psi_1,...,\psi_t : Z^d -> Z
of linear forms all take prime values. The proof is a relatively quick
application of the results in our recent companion paper on the distribution of
polynomial orbits on nilmanifolds.
  We give some applications of our main theorem. We show, for example, that the
Mobius function is uncorrelated with any bracket polynomial. We also obtain a
result about the distribution of nilsequences n -> a^nxL as n ranges only over
the primes.
