A well-known heuristic in number theory is that the Mobius function mu(n) oscillates so randomly that it is asymptotically unbiased with respect to any ”low-complexity” sequence. For instance, the prime number theorem asserts that mu(n) is unbiased with respect to 1, the prime number theorem in arithmetic progressions asserts that mu(n) is unbiased with respect to any periodic sequence, and a classical result of Davenport asserts that mu(n) is unbiased with respect to any almost periodic sequence. The Mobiusnilsequences conjecture asserts that mu(n) is in fact unbiased with respect to any nilsequence - a sequence arising from a flow on a nilpotent manifold. This conjecture, if true, would (in conjunction with another conjecture, the Inverse Conjecture for the Gowers

norm) imply a general asymptotic for the number of solutions of (finite complexity) systems of linear equations with prime variables. Recently, Ben Green and I have established the Mobius-nilsequences conjecture, by combining a classical method of Vaughan with an equidistribution result for nilsequences. We will sketch the proof of this conjecture and its connections with linear equations of primes in this talk.