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.
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.
Keywords: Quadratic uniformity, Möbius function
Mot clés : uniformité quadratique, fonction de Möbius
Green, Ben 1; Tao, Terence 2
@article{AIF_2008__58_6_1863_0, 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{\textquoteright}institut Fourier}, volume = {58}, number = {6}, year = {2008}, doi = {10.5802/aif.2401}, mrnumber = {2473624}, zbl = {1160.11017}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2401/} }
TY - JOUR AU - Green, Ben AU - Tao, Terence TI - Quadratic uniformity of the Möbius function JO - Annales de l'Institut Fourier PY - 2008 SP - 1863 EP - 1935 VL - 58 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2401/ DO - 10.5802/aif.2401 LA - en ID - AIF_2008__58_6_1863_0 ER -
%0 Journal Article %A Green, Ben %A Tao, Terence %T Quadratic uniformity of the Möbius function %J Annales de l'Institut Fourier %D 2008 %P 1863-1935 %V 58 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2401/ %R 10.5802/aif.2401 %G en %F AIF_2008__58_6_1863_0
Green, Ben; Tao, Terence. Quadratic uniformity of the Möbius function. Annales de l'Institut Fourier, Volume 58 (2008) no. 6, pp. 1863-1935. doi : 10.5802/aif.2401. https://aif.centre-mersenne.org/articles/10.5802/aif.2401/
[1] 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
[2] Structure of sets with small sumset, Astérisque (1999) no. 258, pp. xi, 77-108 (Structure theory of set addition) | MR | Zbl
[3] 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 | Zbl
[4] On -subsets of squares, Israel J. Math., Volume 67 (1989) no. 3, pp. 291-311 | DOI | MR | Zbl
[5] Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, 18, Cambridge University Press, Cambridge, 1990 (Basic theory and examples) | MR | Zbl
[6] On some infinite series involving arithmetical functions. II, Quart. J. Math. Oxf., Volume 8 (1937), pp. 313-320 | DOI
[7] Multiplicative number theory, Graduate Texts in Mathematics, 74, Springer-Verlag, New York, 2000 (Revised and with a preface by Hugh L. Montgomery) | MR | Zbl
[8] 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 | Zbl
[9] A new proof of Szemerédi’s theorem, Geom. Funct. Anal., Volume 11 (2001) no. 3, pp. 465-588 | DOI | Zbl
[10] Linear equations in primes (to appear in Annals of Math) | Zbl
[11] An inverse theorem for the Gowers -norm, Proc. Edinburgh Math. Soc., Volume 51 (2008) no. 1, pp. 73-153 | DOI | MR | Zbl
[12] The primes contain arbitrarily long arithmetic progressions, Annals of Math., Volume 167 (2008), pp. 481-547 | DOI | MR | Zbl
[13] 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 | Zbl
[14] Some results in the additive prime number theory, Quart. J. Math. Oxford, Volume 9 (1938), pp. 68-80 | DOI | JFM | Zbl
[15] Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004 | MR | Zbl
[16] On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., Volume 13 (1949), pp. 9-32
[17] Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994 | MR | Zbl
[18] On an additive property of squares and primes, Acta Arith., Volume 49 (1988) no. 3, pp. 281-289 | EuDML | MR | Zbl
[19] Arithmetic progressions and the primes, Collect. Math. (2006) no. Vol. Extra, pp. 37-88 | EuDML | MR | Zbl
[20] Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006 | MR | Zbl
[21] The Hardy-Littlewood method, Cambridge Tracts in Mathematics, 125, Cambridge University Press, Cambridge, 1997 | MR | Zbl
[22] Sommes trigonométriques sur les nombres premiers, C. R. Acad. Sci. Paris Sér. A-B, Volume 285 (1977) no. 16, p. A981-A983 | MR | Zbl
[23] Some theorems concerning the primes, Mat. Sbornik. N.S., Volume 2 (1937), pp. 179-195 | EuDML | Zbl
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