The Möbius function is strongly orthogonal to nilsequences

Ben Green¹ ; Terence Tao²

¹ Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, England
² Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1596

Annals of mathematics, Tome 175 (2012) no. 2, pp. 541-566.

Résumé

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.

Détail

DOI : 10.4007/annals.2012.175.2.3