The Möbius function is strongly orthogonal to nilsequences

Abstract

We show that the Mobius function (n) is strongly asymptotically or- thogonal to any polynomial nilsequence (F (g(n))) n2N. Here, G is a sim- ply-connected nilpotent Lie group with a discrete and cocompact subgroup (so G= is a nilmanifold), g : Z ! G is a polynomial sequence, and F : G= ! R is a Lipschitz function. More precisely, we show that j 1 N P N n=1 (n)F (g(n)) j F;G; ;A log A N for all A > 0. In particu- lar, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major in- gredients in our programme to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collec- tion 1;:::; 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. We give some applications of our main theorem. We show, for exam- ple, that the Mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c. We also obtain a result about the distribution of nilse- quences (a n x) n2N as n ranges only over the primes.