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Quadratic uniformity of the Möbius function. (English) Zbl 1160.11017
Since proving their celebrated theorem on arithmetic progressions in the primes [Ann. Math. (2) 171, No. 3, 1753–1850 (2010; Zbl 1242.11071)] the authors have embarked on a program to handle the solution of a wide class of systems of linear equations in the primes. The program has two components, the so-called Gowers inverse conjectures, and the Möbius nilsequence conjectures, and the paper under review establishes the first new case of the latter.
The motivation behind the paper is the metaprinciple that the Möbius function behaves sufficiently randomly that it is asymptotically orthogonal to any low complexity object. Our classical knowledge in this regard may be reformulated in the language of the paper as the Möbius nilsequence conjecture (theorem) for 1-step nilsequences, which we record now from Example 4 of the paper.
Suppose that $$G$$ is a connected, simply-connected abelian Lie group with a smooth metric $$d$$, and that $$\Gamma$$ is a closed cocompact subgroup of $$G$$. Then $$G/\Gamma$$ is called a $$1$$-step nilmanifold; it is a torus. Furthermore suppose that $$F:G/\Gamma \rightarrow \mathbb C$$ is a Lipschitz function of constant $$1$$, and write $$T_g:G/\Gamma \rightarrow G/\Gamma$$ for left translation by $$g$$ on $$G/\Gamma$$. Then one has the estimate $\mathbf E_{n \in [N]}{\mu(n)\overline{F(T_g^nx)}} = O_{A,G/\Gamma}(\log^{-A}N)$ for all $$A>0$$, integers $$N\geq 1$$, elements $$g \in G$$ and $$x \in G/\Gamma$$. In the paper the authors establish this in the more general case when $$G/\Gamma$$ is a $$2$$-step nilmanifold. The proof is long but written in a very accessible style and many of the ingredients will be recognized by those having some familiarity with analytic number theory. Comprehensive appendices make the paper essentially self-contained.

##### MSC:
 11B99 Sequences and sets
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