[Paper Review] Quadratic Uniformity of the Mobius Function

Ben Green, Terence Tao|arXiv (Cornell University)|Jun 4, 2006

Analytic Number Theory Research11 references4 citations

TL;DR

This paper establishes the Möbius and Nilsequences Conjecture for 1- and 2-step nilsequences, proving that the Möbius function is strongly asymptotically orthogonal to Lipschitz functions on such nilmanifolds. The result generalizes Davenport's exponential sum estimate and provides a key ingredient for asymptotic formulas in prime number theory, including four-term prime progressions and non-degenerate affine lattice problems of codimension at most two.

ABSTRACT

This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In particular, the results of this paper may be used, together with the machinery of [LEP], to establish an asymptotic for the number of four-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally any problem counting prime points inside a ``non-degenerate'' affine lattice of codimension at most 2. The main result of this paper is a proof of the Mobius and Nilsequences Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in [LEP] and amounts to showing that if G/Γis an s-step nilmanifold, s <= 2, if F : G/Γ-> [-1,1] is a Lipschitz function, and if T_g : G/Γ-> G/Γis the action of g \in G on G/Γ, then the Mobius function μ(n) is orthogonal to the sequence F(T_g^n x) in a fairly strong sense, uniformly in g and x in G/Γ. This can be viewed as a ``quadratic'' generalisation of an exponential sum estimate of Davenport, and is proven by the following the methods of Vinogradov and Vaughan.

Motivation & Objective

• To prove the Möbius and Nilsequences Conjecture for 1- and 2-step nilmanifolds, extending Davenport's classical exponential sum estimate.
• To establish strong asymptotic orthogonality between the Möbius function and Lipschitz functions on nilmanifolds, with bounds decaying faster than any power of log N.
• To provide a foundational tool for the Hardy-Littlewood method in systems of linear equations in primes, particularly for problems of codimension at most two.
• To generalize exponential sum techniques of Vinogradov and Vaughan to the nilsequence setting, enabling the analysis of higher-order correlations in the Möbius function.
• To support the asymptotic counting of prime points in non-degenerate affine lattices, such as four-term arithmetic progressions of primes.

Proposed method

• Adapts the methods of Vinogradov and Vaughan to handle quadratic exponential sums over nilmanifolds, using major and minor arc decompositions.
• Employs a coordinate transformation to express nilsequences in terms of quadratic forms and fractional parts, reducing the problem to estimating oscillatory sums.
• Applies the theory of 2-step nilpotent Lie groups and their homogeneous spaces to model the dynamics of nilrotations on nilmanifolds.
• Uses the concept of Lipschitz functions on nilmanifolds and their equidistribution properties to control the correlation with the Möbius function.
• Applies divisor moment estimates (e.g., $\mathbb{E}_{n\leq N} \tau(n)^m \ll_m (\log N)^{2^m - 1}$) to control collision probabilities in sieve-theoretic arguments.
• Combines the main term estimates with the divisor packing lemma to control the number of overlapping solutions in arithmetic configurations.

Experimental results

Research questions

• RQ1 Can the Möbius function be shown to be strongly asymptotically orthogonal to 2-step nilsequences, generalizing Davenport's result for linear phases?
• RQ2 What is the quantitative decay rate of the correlation $\mathbb{E}_{n\leq N} \mu(n) F(T_g^n \cdot x)$ for Lipschitz functions on 1- and 2-step nilmanifolds?
• RQ3 How can the Hardy-Littlewood method be extended to systems of linear equations in primes using nilsequence orthogonality?
• RQ4 What is the role of nilpotent group dynamics in controlling the cancellation in multiplicative functions over polynomial sequences?
• RQ5 Can the Möbius function's cancellation be quantified uniformly over nilrotations, with error terms decaying faster than any power of log N?

Key findings

• The Möbius function is strongly asymptotically orthogonal to 1- and 2-step nilsequences: $\left| \mathbb{E}_{n\leq N} \mu(n) F(T_g^n \cdot x) \right| \ll_A \|F\|_{\text{Lip}} \log^{-A} N$ uniformly in $g \in G$ and $x \in G/\Gamma$.
• The result extends Davenport's estimate $\mathbb{E}_{n\leq N} \mu(n) e(-\alpha n) \ll_A \log^{-A} N$ to quadratic phases and nilsequences.
• The proof relies on a decomposition into major and minor arcs, with the minor arc contribution controlled via Vinogradov-type estimates.
• The divisor packing lemma $|\bigcup_{d\in\mathfrak{D}} A_d| \gg_\kappa \delta^2 |\mathfrak{D}|^2 |A| \alpha^\kappa \log^{-2^{2/\kappa}} N$ is used to control collisions in sieve-theoretic settings.
• The main result enables the asymptotic counting of four-term prime progressions $p_1 < p_2 < p_3 < p_4 \leq N$ and more general prime points in non-degenerate affine lattices of codimension ≤ 2.
• The implied constants in the bounds are ineffective due to potential Landau-Siegel zeros, consistent with classical results in sieve theory.