The Mobius function is strongly orthogonal to nilsequences

TL;DR

This paper proves that the Mobius function is strongly orthogonal to polynomial nilsequences, confirming a key conjecture and advancing understanding of the distribution of primes and nilsequences in number theory.

Contribution

It establishes the Mobius and Nilsequence conjecture MN(s) for all positive integers s, linking the orthogonality of Mobius to nilsequences with prime number distribution.

Findings

Mobius function is uncorrelated with bracket polynomials.

Mobius function shows strong orthogonality to polynomial nilsequences.

Results support the generalized Hardy-Littlewood conjecture for primes.

Abstract

We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a Lipschitz function. More precisely, we show that the inner product of mu(n) with F(g(n)L) over {1,...,N} is bounded by 1/log^A N, for all A > 0. In particular, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper "Linear equations in primes" for every positive integer s. This is one of two major ingredients in our programme, outlined in that paper, 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 on the distribution of polynomial orbits on nilmanifolds. We give some applications of our main theorem. We show, for example, that the Mobius function is uncorrelated with any bracket polynomial. We also obtain a result about the distribution of nilsequences n -> a^nxL as n ranges only over the primes.