The Mobius function is strongly orthogonal to nilsequences

By Ben Joseph Green, Terence Tao. 2008

269 citations recorded by the source corpus.

We show that the Mobius function (n) is strongly asymptotically or- thogonal to any polynomial nilsequence (F (g(n))) n2N. Here, G is a sim- ply-connected nilpotent Lie group with a discrete and cocompact subgroup (so G= is a nilmanifold), g : Z ! G is a polynomial sequence, and F : G= ! R is a Lipschitz function. More precisely, we show that j 1 N P N n=1 (n)F (g(n)) j F;G; ;A log A N for all A > 0. In particu- lar, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major in- gredients in our programme to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collec- tion 1;:::; 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 exam- ple, that the Mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c. We also obtain a result about the distribution of nilse- quences (a n x) n2N as n ranges only over the primes.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Ophthalmology

View Paper: https://www.semanticscholar.org/paper/3b28fc6ba8f4a424aa27ede8ef218e1d8f7a30ac

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Citations in the corpus

Higher order Fourier analysis of multiplicative functions and applications (2014, methodology): “The present statement follows immediately from the result in [24] and Lemma A.”
Prime numbers along Rudin–Shapiro sequences (2015, background): “an asymptotic formula for the more difficult to handle sum ∑ n≤x Λ(n)un), see [6], [7], [12], [20].”
Sarnak’s Conjecture: What’s New (2017, methodology): “Green and Tao in [80] proved Möbius disjointnes in the following strong form: Theorem 5.”
The logarithmic Sarnak conjecture for ergodic weights (2017, background): “4 in various places) the Gowers uniformity of the W -tricked von Mangoldt function which was established in [32, 34, 35].”
Odometers and Toeplitz systems revisited in the context of Sarnak's conjecture (2015, result): “t ˘(n) = f(Tnx) (for n1). Then ˘and are uncorrelated. The conjecture is known to hold for relatively few types of dynamical systems, in particular for odometers, irrational rotations, nilsequences [GT], horocycle ows [BSZ]. See also [B1], [B], [ALR], [G], [KL], [LS], [MR], for other results. We rem”
The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures (2017, background): “It is only known that the Liouville function is Gowers uniform over long intervals (by [19], [20]), as opposed to short ones, and even the case of local U(2)-uniformity of the Liouville function is a difficult open problem (also mentioned in [36, Section 4]).”
Measure complexity and Möbius disjointness (2017, background): “[30] K. Matomäki, M. Radziwi l l, and T. Tao, An averaged form of Chowla’s conjecture, Algebra Number Theory 9 (2015), no. 9, 2167-2196.”
Disjointness of Moebius from Horocycle Flows (2013, methodology): “ deterministic systems. For (X,T) a Kronecker ﬂow (that is a translation in a compact abelian group) it is proven in [V] and [D] while for (X,T) a translation on a compact nilmanifold it is proved in [G-T]. It is also known for some substitution dynamics associated with the Morse sequence [M-R].(∗) ”
Equivalence of the Logarithmically Averaged Chowla and Sarnak Conjectures (2016, background): “CHOWLA AND SARNAK CONJECTURES 3 dynamical systems [1], [2], [3], [5], [6], [7], [8], [9], [12], [13], [14], [15], [17], [18], [25], [29], [33], [39], [40], [41], [42], [43], [48], [49], [52], [58]; for further variants of the Sarnak conjecture, see [10], [13], [37].”
The horocycle flow at prime times (2011, unknown): “N T ] 2 J un 2 01 3”
The Möbius function and distal flows (2013, methodology): “1 of [9] using the constructions with disconnnected nilmanifolds as in [15].”
Some open problems on multiple ergodic averages (2011, background): “such equidistribution problems have been developed in recent years, thus making such a reduction very much worthwhile. Some examples of useful equidistribution results on nilmanifolds can be found in [17, 130, 131, 141, 169, 170, 171, 185, 227, 228, 230, 231, 233, 241, 242, 299]. 2.6. A general stra”