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The Möbius function is strongly orthogonal to nilsequences. (English) Zbl 1347.37019

This paper is a part of the Green-Tao program to establish asymptotic results for the number of solutions of systems of linear equations under the assumption that all variables are assigned prime values. A landmark paper in this direction was [Ann. Math. (2) 167, No. 2, 481–547 (2008; Zbl 1191.11025)] where the authors showed that the sequence of primes contains arbitrarily long arithmetic progressions. In [the authors, Ann. Math. (2) 171, No. 3, 1753–1850 (2010; Zbl 1242.11071)] they proved a general theorem including asymptotic formulas for the number of solutions, conditionally on the truth of two conjectures MN(s) and GI(s). Conjecture GI(s) was proved by the first author et al. [Ann. Math. (2) 176, No. 2, 1231–1372 (2012; Zbl 1282.11007)], while the present paper contains (as one application) the proof of conjecture MN(s) for \(s \geq 3\). Conjecture MN(2) had already been proved by the authors [Ann. Inst. Fourier 58, No. 6, 1863–1935 (2008; Zbl 1160.11017)]. As a further remark, the present paper has a companion paper [Zbl 1251.37012] which is longer and more technical, and which establishes several deep results that are used here (mainly quantitative equidistribution results for polynomial orbits on nilmanifolds).
The main result of the paper is the following. Let \(G\) be a simply connected nilpotent Lie group which has a discrete and cocompact subgroup \(\Gamma\), meaning that \(G / \Gamma\) is a nilmanifold. Let \(g: \mathbb{Z} \mapsto G\) be a polynomial sequence, and let \(F:~G/\Gamma \mapsto \mathbb{R}\) be a Lipschitz function. Then the authors prove that \[ \left| N^{-1} \sum_{n=1}^N \mu(n) F(g(n) \Gamma) \right| \ll_{F,G,\Gamma,A} \log^{-A} N \] for all positive \(A\). This quantitative result goes beyond the previously known qualitative results from [J. Bourgain et al., Dev. Math. 28, 67–83 (2013; Zbl 1336.37030)]. The proof uses a classical decomposition into “Type I” and “Type II” sums, a method going back to Vinogradov.
Further applications of the methods developed in this paper are:
a) an orthogonality result between the Möbius function and bracket polynomials, which is a somewhat natural supplement to the main theorem due to the close relation between bracket polynomials and nilmanifolds shown by V. Bergelson and A. Leibman [Acta Math. 198, No. 2, 155–230 (2007; Zbl 1137.37005)].
b) corresponding results when the Möbius function \(\mu\) is replaced by the Liouville function \(\lambda\).
c) a recurrence/convergence theorem along the primes.

MSC:

37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11N13 Primes in congruence classes
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References:

[1] V. Bergelson and I. J. Håland, ”Sets of recurrence and generalized polynomials,” in Convergence in Ergodic Theory and Probability, Berlin: de Gruyter, 1996, vol. 5, pp. 91-110. · Zbl 0958.28014
[2] V. Bergelson and A. Leibman, ”Distribution of values of bounded generalized polynomials,” Acta Math., vol. 198, iss. 2, pp. 155-230, 2007. · Zbl 1137.37005 · doi:10.1007/s11511-007-0015-y
[3] H. Davenport, ”On some infinite series involving arithmetical functions. II,” Quart. J. Math. Oxford, vol. 8, pp. 313-320, 1937. · Zbl 0017.39101
[4] B. Green, ”Generalising the Hardy-Littlewood method for primes,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2007, pp. 373-399. · Zbl 1157.11007
[5] B. Green, ”Three topics in additive prime number theory,” in Current Developments in Mathematics, 2007, Int. Press, Somerville, MA, 2009, pp. 1-41. · Zbl 1230.11113
[6] B. Green and T. Tao, ”An inverse theorem for the Gowers \(U^3(G)\) norm,” Proc. Edinb. Math. Soc., vol. 51, iss. 1, pp. 73-153, 2008. · Zbl 1202.11013 · doi:10.1017/S0013091505000325
[7] B. Green and T. Tao, ”Quadratic uniformity of the Möbius function,” Ann. Inst. Fourier \((\)Grenoble\()\), vol. 58, iss. 6, pp. 1863-1935, 2008. · Zbl 1160.11017 · doi:10.5802/aif.2401
[8] B. Green and T. Tao, ”Linear equations in primes,” Ann. of Math., vol. 171, pp. 1753-1850, 2010. · Zbl 1242.11071 · doi:10.4007/annals.2010.171.1753
[9] B. Green and T. Tao, ”The quantitative behaviour of polynomial orbits on nilmanifolds,” Ann. of Math., vol. 175, pp. 465-540, 2012. · Zbl 1251.37012 · doi:10.4007/annals.2012.175.2.2
[10] I. J. Håland, ”Uniform distribution of generalized polynomials,” J. Number Theory, vol. 45, iss. 3, pp. 327-366, 1993. · Zbl 0797.11064 · doi:10.1006/jnth.1993.1082
[11] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53. · Zbl 1071.11001
[12] A. Leibman, ”Polynomial sequences in groups,” J. Algebra, vol. 201, iss. 1, pp. 189-206, 1998. · Zbl 0908.20029 · doi:10.1006/jabr.1997.7269
[13] A. Leibman, ”Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold,” Ergodic Theory Dynam. Systems, vol. 25, iss. 1, pp. 201-213, 2005. · Zbl 1080.37003 · doi:10.1017/S0143385704000215
[14] A. I. Mal\('\)cev, ”On a class of homogeneous spaces,” Izvestiya Akad. Nauk. SSSR. Ser. Mat., vol. 13, pp. 9-32, 1949.
[15] T. Tao, ”Obstructions to uniformity and arithmetic patterns in the primes,” Pure Appl. Math. Q., vol. 2, iss. 2, part 2, pp. 395-433, 2006. · Zbl 1105.11032 · doi:10.4310/PAMQ.2006.v2.n2.a2
[16] T. Tao, ”The dichotomy between structure and randomness, arithmetic progressions, and the primes,” in International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, 2007, pp. 581-608. · Zbl 1183.11008 · doi:10.4171/022-1/22
[17] R. C. Vaughan, ”Sommes trigonométriques sur les nombres premiers,” C. R. Acad. Sci. Paris Sér. A-B, vol. 285, iss. 16, p. a981-a983, 1977. · Zbl 0374.10025
[18] R. C. Vaughan, The Hardy-Littlewood Method, Second ed., Cambridge: Cambridge Univ. Press, 1997, vol. 125. · Zbl 0868.11046
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