The Mobius function is strongly orthogonal to nilsequences

• 成果类型: Article
• 署名作者: Green, Ben; Tao, Terence
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.175.2.3
• 发表日期: 2012
• 页码: 541-566
• 关键词: generalized polynomials; UNIFORMITY; primes

摘要

We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F(g(n)Gamma))(n is an element of N). Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup F (so G/Gamma is a nilmanifold), g : Z -> G is a polynomial sequence, and F : G/Gamma -> R is a Lipschitz function. More precisely, we show that vertical bar 1/N Sigma(N)(n=1),mu(n)F(g(n)Gamma)vertical bar <<(F,G,Gamma,A) log(-A) N for all A > 0. In particular, this implies the Mobius and Nilseguence conjecture 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), . . . , 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. We give some applications of our main theorem. We show, for example, that the Mobius function is uncorrelated with any bracket polynomial such as n root 3left perpendicularn root 2right perpendicular. We also obtain a result about the distribution of nilsequences (a(n)x Gamma)(n is an element of N) as n ranges only over the primes.