The Möbius function is strongly orthogonal to nilsequences


We show that the Möbius function $\mu(n)$ is strongly asymptotically orthogonal to any polynomial nilsequence $(F(g(n)\Gamma))_{n \in \mathbb{N}}$. Here, $G$ is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup $\Gamma$ (so $G/\Gamma$ is a nilmanifold), $g : \mathbb{Z} \rightarrow G$ is a polynomial sequence, and $F: G/\Gamma \to \Bbb{R}$ is a Lipschitz function. More precisely, we show that $|\frac{1}{N} \sum_{n=1}^N \mu(n) F(g(n) \Gamma)| \ll_{F,G,\Gamma,A} \log^{-A} N$ for all $A > 0$. In particular, this implies the Möbius and Nilsequence conjecture $\mbox{MN}(s)$ from our earlier paper for every positive integer $s$. This is one of two major ingredients in our programme to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection $\psi_1,\dots,\psi_t : \mathbb{Z}^d \rightarrow \mathbb{Z}$ 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 example, that the Möbius function is uncorrelated with any bracket polynomial such as $n\sqrt{3}\lfloor n\sqrt{2}\rfloor$. We also obtain a result about the distribution of nilsequences $(a^nx\Gamma)_{n \in \mathbb{N}}$ as $n$ ranges only over the primes.


Ben Green

Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, England

Terence Tao

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1596