Quadratic uniformity of the Möbius function  [ L’uniformité quadratique de la fonction de Möbius ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 1863-1935.

On établit la conjecture « Möbius et Nilsuites » pour les nilsystèmes de rang 1 et 2. Ce papier est une partie de notre programme, dont le but est une généralisation de la méthode de Hardy-Littlewood en vue d’étudier les systèmes d’équations linéaires dans les nombres premiers.

We prove the “Möbius and Nilsequences Conjecture” for nilsystems of step 1 and 2. This paper forms a part of our program to generalise the Hardy-Littlewood method so as to handle systems of linear equations in primes.

DOI : https://doi.org/10.5802/aif.2401
Classification : 11B99
Mots clés : uniformité quadratique, fonction de Möbius
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     author = {Green, Ben and Tao, Terence},
     title = {Quadratic uniformity of the M\"obius function},
     journal = {Annales de l'Institut Fourier},
     pages = {1863--1935},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {6},
     year = {2008},
     doi = {10.5802/aif.2401},
     mrnumber = {2473624},
     zbl = {1160.11017},
     language = {en},
     url = {www.numdam.org/item/AIF_2008__58_6_1863_0/}
}
Green, Ben; Tao, Terence. Quadratic uniformity of the Möbius function. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 1863-1935. doi : 10.5802/aif.2401. http://www.numdam.org/item/AIF_2008__58_6_1863_0/

[1] Auslander, L.; Green, L.; Hahn, F. Flows on homogeneous spaces, With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53, Princeton University Press, Princeton, N.J., 1963 | Zbl 0106.36802

[2] Bilu, Yuri Structure of sets with small sumset, Astérisque (1999) no. 258, pp. xi, 77-108 (Structure theory of set addition) | MR 1701189 | Zbl 0946.11004

[3] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998 (Translated from the French, Reprint of the 1989 English translation) | MR 1728312 | Zbl 0672.22001

[4] Bourgain, J. On Λ(p)-subsets of squares, Israel J. Math., Volume 67 (1989) no. 3, pp. 291-311 | Article | MR 1029904 | Zbl 0692.43005

[5] Corwin, Lawrence J.; Greenleaf, Frederick P. Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, Volume 18, Cambridge University Press, Cambridge, 1990 (Basic theory and examples) | MR 1070979 | Zbl 0704.22007

[6] Davenport, H. On some infinite series involving arithmetical functions. II, Quart. J. Math. Oxf., Volume 8 (1937), pp. 313-320 | Article

[7] Davenport, Harold Multiplicative number theory, Graduate Texts in Mathematics, Volume 74, Springer-Verlag, New York, 2000 (Revised and with a preface by Hugh L. Montgomery) | MR 1790423 | Zbl 1002.11001

[8] Furstenberg, Hillel Nonconventional ergodic averages, The legacy of John von Neumann (Hempstead, NY, 1988) (Proc. Sympos. Pure Math.) Volume 50, Amer. Math. Soc., Providence, RI, 1990, pp. 43-56 | MR 1067751 | Zbl 0711.28006

[9] Gowers, W. T. A new proof of Szemerédi’s theorem, Geom. Funct. Anal., Volume 11 (2001) no. 3, pp. 465-588 | Article | Zbl 1028.11005

[10] Green, B. J.; Tao, T. C. Linear equations in primes (to appear in Annals of Math) | Zbl 1242.11071

[11] Green, B. J.; Tao, T. C. An inverse theorem for the Gowers U 3 -norm, Proc. Edinburgh Math. Soc., Volume 51 (2008) no. 1, pp. 73-153 | Article | MR 2391635 | Zbl 1202.11013

[12] Green, B. J.; Tao, T. C. The primes contain arbitrarily long arithmetic progressions, Annals of Math., Volume 167 (2008), pp. 481-547 | Article | MR 2415379 | Zbl 1191.11025

[13] Green, Ben Finite field models in additive combinatorics, Surveys in combinatorics 2005 (London Math. Soc. Lecture Note Ser.) Volume 327, Cambridge Univ. Press, Cambridge, 2005, pp. 1-27 | MR 2187732 | Zbl 1155.11306

[14] Hua, L. K. Some results in the additive prime number theory, Quart. J. Math. Oxford, Volume 9 (1938), pp. 68-80 | Article | JFM 64.0131.02 | Zbl 0018.29404

[15] Iwaniec, Henryk; Kowalski, Emmanuel Analytic number theory, American Mathematical Society Colloquium Publications, Volume 53, American Mathematical Society, Providence, RI, 2004 | MR 2061214 | Zbl 1059.11001

[16] Mal’cev, A. I. On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., Volume 13 (1949), pp. 9-32

[17] Montgomery, Hugh L. Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, Volume 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994 | MR 1297543 | Zbl 0814.11001

[18] Ruzsa, Imre Z. On an additive property of squares and primes, Acta Arith., Volume 49 (1988) no. 3, pp. 281-289 | EuDML 206086 | MR 932527 | Zbl 0636.10042

[19] Tao, Terence Arithmetic progressions and the primes, Collect. Math. (2006) no. Vol. Extra, pp. 37-88 | EuDML 41790 | MR 2264205 | Zbl 1109.11043

[20] Tao, Terence; Vu, Van Additive combinatorics, Cambridge Studies in Advanced Mathematics, Volume 105, Cambridge University Press, Cambridge, 2006 | MR 2289012 | Zbl 1127.11002

[21] Vaughan, R. C. The Hardy-Littlewood method, Cambridge Tracts in Mathematics, Volume 125, Cambridge University Press, Cambridge, 1997 | MR 1435742 | Zbl 0868.11046

[22] Vaughan, Robert-C. Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris Sér. A-B, Volume 285 (1977) no. 16, p. A981-A983 | MR 498434 | Zbl 0374.10025

[23] Vinogradov, I. M. Some theorems concerning the primes, Mat. Sbornik. N.S., Volume 2 (1937), pp. 179-195 | EuDML 64898 | Zbl 0017.19803