We discuss the notion of a “Level of Distribution” in two settings. The first deals with primes in progressions, and the role this plays in Yitang Zhang’s theorem on bounded gaps between primes. The second concerns the Affine Sieve and its applications.
Nous discutons de la notion de « Niveau de Distribution » dans deux contextes. Le premier concerne les nombres premiers en progression, et le rôle qu’elle joue dans le théorème de Yitang Zhang sur les écarts bornés entre nombres premiers. Le second concerne le Crible Affine et ses applications.
@article{AFST_2014_6_23_5_933_0, author = {Alex Kontorovich}, title = {Levels of {Distribution} and the {Affine} {Sieve}}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {933--966}, publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 23}, number = {5}, year = {2014}, doi = {10.5802/afst.1432}, zbl = {1335.11078}, mrnumber = {3294598}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1432/} }
TY - JOUR AU - Alex Kontorovich TI - Levels of Distribution and the Affine Sieve JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2014 SP - 933 EP - 966 VL - 23 IS - 5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1432/ DO - 10.5802/afst.1432 LA - en ID - AFST_2014_6_23_5_933_0 ER -
%0 Journal Article %A Alex Kontorovich %T Levels of Distribution and the Affine Sieve %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2014 %P 933-966 %V 23 %N 5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1432/ %R 10.5802/afst.1432 %G en %F AFST_2014_6_23_5_933_0
Alex Kontorovich. Levels of Distribution and the Affine Sieve. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 5, pp. 933-966. doi : 10.5802/afst.1432. https://afst.centre-mersenne.org/articles/10.5802/afst.1432/
[1] Barban (M. B.).— The sieve" method and its application to number theory. Uspehi Mat. Nauk, 21, p. 51-102 (1966). | MR | Zbl
[2] Blomer (V.) and Brumley (F.).— On the Ramanujan conjecture over number fields. Ann. of Math. (2), 174(1), p. 581-605 (2011). | MR
[3] Blomer (V.) and Brumley (F.).— The role of the Ramanujan conjecture in analytic number theory. Bull. Amer. Math. Soc. (N.S.), 50(2), p. 267-320 (2013). | MR
[4] Bombieri (E.) and Davenport (H.).— Small differences between consecutive prime numbers. Proc. Roy. Soc. Ser. A, p. 1-18 (1966). | MR | Zbl
[5] Bombieri (E.), Friedlander (J.), and Iwaniec (H.).— Primes in arithmetic progressions to large moduli. Acta Math., 156, p. 203-251 (1986). | MR | Zbl
[6] Bourgain (J.) and Gamburd (A.).— Uniform expansion bounds for Cayley graphs of SL2( Fp). Ann. of Math. (2), 167(2), p. 625-642 (2008). | MR | Zbl
[7] Bourgain (J.), Gamburd (A.), and Sarnak (P.).— Sieving and expanders. C. R. Math. Acad. Sci. Paris, 343(3), p. 155-159 (2006). | MR | Zbl
[8] Bourgain (J.), Gamburd (A.), and Sarnak (P.).— Affine linear sieve, expanders, and sum-product. Invent. Math., 179(3), p. 559-644 (2010). | MR | Zbl
[9] Bourgain (J.), Gamburd (A.), and Sarnak (P.).— Generalization of Selberg’s 3/16th theorem and affine sieve. Acta Math, 207, p. 255-290 (2011). | MR | Zbl
[10] Breuillard (E.), Green (B.), and Tao (T.).— Approximate subgroups of linear groups. Geom. Funct. Anal., 21(4), p. 774-819 (2011). | MR | Zbl
[11] Bourgain (J.) and Kontorovich (A.).— On Zaremba’s conjecture.— Comptes Rendus Mathematique, 349(9), p. 493-495 (2011). | MR | Zbl
[12] Bourgain (J.) and Kontorovich (A.).— On the local-global conjecture for integral Apollonian gaskets (2012). To appear, Invent. Math., arXiv:1205.4416v1, 63 p. 27. | MR
[13] Bourgain (J.) and Kontorovich (A.).— The affine sieve beyond expansion I: thin hypotenuses (2013). Preprint, arXiv:1307.3535. | MR
[14] Bourgain (J.) and Kontorovich (A.).— Beyond expansion II: Traces of thin semigroups (2013). Preprint, arXiv:1310.7190.
[15] Bourgain (J.) and Kontorovich (A.).— On Zaremba’s conjecture. Annals Math., 180(1), p. 137-196 (2014). | MR
[16] Bombieri (E.).— On the large sieve. Mathematika, 12, p. 201-225 (1965). | MR | Zbl
[17] Bourgain (J.).— Integral Apollonian circle packings and prime curvatures. J. Anal. Math., 118(1), p. 221-249 (2012). | MR | Zbl
[18] Brun (V.).— Le crible d’Eratosthéne et le theoréme de Goldbach. C. R. Acad. Sci. Paris, 168, p. 544-546 (1919).
[19] Burger (M.) and Sarnak (P.).— Ramanujan duals II. Invent. Math, 106, p. 1-11 (1991). | MR | Zbl
[20] Chen (J. R.).— On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica, 16, p. 157-176 (1973). | MR | Zbl
[21] Clozel (L.).— Démonstration de la conjecture . Invent. Math., 151(2), p. 297-328 (2003). | MR | Zbl
[22] Daveport (H.).— Multiplicative Number Theory, volume 74 of Grad. Texts Math. Springer-Verlag, New York (1980). | MR | Zbl
[23] Deuring (M.).— Imaginäre quadratische Zahlkörper mit der Klassenzahl 1. Math. Z., 37(1), p. 405-415 (1933). | MR | Zbl
[24] de la Vallée-Poussin (Ch.J.).— Recherches analytiques sur la théorie des nombers premiers. Ann. Soc. Sci. Bruxelles, 20, p. 183-256 (1896).
[25] Elliott (P.D.T.A.) and Halberstam (H.).— A conjecture in prime number theory. Symp. Math. IV (Rome 1968/69), p. 59-72 (1968). | MR | Zbl
[26] Erdös (P.).— The difference between consecutive primes. Duke Math J., 6, p. 438-441 (1940). | MR | Zbl
[27] Friedlander (J.) and Granville (A.).— Limitations to the equidistribution of primes. I. Ann. of Math. (2), 129(2), p. 363-382 (1989). | MR | Zbl
[28] Fouvry (E.) and Iwaniec (H.).— Primes in arithmetic progressions. Acta Arith., 42, p. 197-218 (1983). | MR | Zbl
[29] Friedlander (J.) and Iwaniec (H.).— What is ... the parity phenomenon? Notices Amer. Math. Soc., 56(7), p. 817-818 (2009). | MR | Zbl
[30] Friedlander (J.) and Iwaniec (H.).— Hyperbolic prime number theorem. Acta Math., 202(1), p. 1-19 (2009). | MR | Zbl
[31] Friedlander (J.) and Iwaniec (H.).— Opera de cribro, volume 57 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (2010). | MR | Zbl
[32] Friedlander (J.) and Iwaniec (H.).— Close encounters among the primes (2014). arXiv:1312.2926. | MR
[33] Frolenkov (D.) and Kan (I. D.).— A reinforcement of the Bourgain-Kontorovich’s theorem by elementary methods II (2013). Preprint, arXiv:1303.3968.
[34] Fuchs (E.), Meiri (C.), and Sarnak (P.).— Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, 2012. To appear, JEMS. | MR
[35] Fouvry (E.).— Autour du théorème de Bombieri-Vinogradov. Acta Math, 152, p. 219-244 (1984). | MR | Zbl
[36] Fuchs (E.).— The ubiquity of thin groups (2012). To appear, MSRI Proceedings. | MR
[37] Gamburd (A.).— On the spectral gap for infinite index “congruence" subgroups of SL2(Z). Israel J. Math., 127, p. 157-200 (2002). | MR | Zbl
[38] Goldston (D. A.), Graham (S. W.), Pintz (J.), and Yildirim (C. Y.).— Small gaps between products of two primes. Proc. London Math. Soc., 98(3), p. 741-774 (2009). | MR | Zbl
[39] Goldfeld (D.).— The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3(4), p. 624-663 (1976). | Numdam | MR | Zbl
[40] Goldfeld (D.).— Gauss’s class number problem for imaginary quadratic fields. Bull. Amer. Math. Soc. (N.S.), 13(1), p. 23-37 (1985). | MR | Zbl
[41] Goldston (D. A.).— On Bombieri and Davenport’s theorem concerning small gaps between primes. Mathematika, 39(1), p. 10-17 (1992). | MR | Zbl
[42] Goldston (D. A.), Pintz (J.), and Yildirim (C. Y.).— The path to recent progress on small gaps between primes. In Analytic number theory, volume 7 of Clay Math. Proc., pages 129-139. Amer. Math. Soc., Providence, RI (2007). | MR | Zbl
[43] Goldston (D. A.), Pintz (J.), and Yildirim (C. Y.).— Primes in tuples I. Ann. of Math. (2), 170(2), p. 819-862 (2009). | MR | Zbl
[44] Goldston (D. A.), Pintz (J.), and Yildirim (C. Y.).— Primes in tuples II. Acta Math., 204, p. 1-47 (2010). | MR | Zbl
[45] Granville (A.).— Harald Cramér and the distribution of prime numbers. Scand. Actuar. J., (1), p. 12-28 (1995). Harald Cramér Symposium (Stockholm, 1993). | MR | Zbl
[46] Green (B.).— Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott, and Sarnak. Current Events Bulletin, AMS (2010).
[47] Green (B.) and Tao (T.).— Linear equations in primes. Ann. of Math. (2), 171(3), p. 1753-1850 (2010). | MR | Zbl
[48] Gross (B. H.) and Zagier (D. B).— Heegner points and derivatives of L-series. Invent. Math., 84(2), p. 225-320 (1986). | MR | Zbl
[49] Hadamard (J.).— Sur la distribution des zéros de la fonction et ses conséquences arithmétiques. Bull. Soc. Math. France, 24, p. 199-220 (1896). | MR
[50] Heath-Brown (D. R.).— Prime twins and Siegel zeros. Proc. London Math. Soc. (3), 47(2), p. 193-224 (1983). | MR | Zbl
[51] Heilbronn (H.).— On the class number in imaginary quadratic elds. Quarterly J. of Math., 5, p. 150-160 (1934). | Zbl
[52] Helfgott (H. A.).— Growth and generation in . Ann. of Math. (2), 167(2), p. 601-623 (2008). | MR | Zbl
[53] Hong (J.) and Kontorovich (A.).— Almost prime coordinates for anisotropic and thin Pythagorean orbits (2014). To appear, Israel J. Math. arXiv:1401.4701.
[54] Hoory (S.), Linial (N.), and Wigderson (A.).— Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4), p. 439-561 (electronic), 2006. | MR | Zbl
[55] Huang (S.).— An improvement on Zaremba’s conjecture (2013). Preprint, arXiv:1310.3772.
[56] Huxley (M. N.).— Small differences between consecutive primes. II. Mathematika, 24, p. 142-152 (1977). | MR | Zbl
[57] Iwaniec (H.) and Kowalski (E.).— Analytic number theory, volume 53 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004. | MR | Zbl
[58] Kim (H. H.).— Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. J. Amer. Math. Soc., 16(1), p. 139-183 (electronic) (2003). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Sarnak (P.). | MR | Zbl
[59] Kontorovich (A.) and Oh (H.).— Almost prime Pythagorean triples in thin orbits. J. reine angew. Math., 667, p. 89-131 (2012). arXiv:1001.0370. | MR | Zbl
[60] Kontorovich (A. V.).— The Hyperbolic Lattice Point Count in Infinite Volume with Applications to Sieves. Columbia University Thesis (2007). | MR
[61] Kontorovich (A.).— The hyperbolic lattice point count in infinite volume with applications to sieves. Duke J. Math., 149(1), p. 1-36 (2009). arXiv:0712.1391. | MR | Zbl
[62] Kontorovich (A.).— Expository note: an arithmetic surface (2011). Unpublished note, http://math.yale.edu/ avk23/files/UniformLattice. pdf.
[63] Kontorovich (A.).— From Apollonius to Zaremba: local-global phenomena in thin orbits. Bull. Amer. Math. Soc. (N.S.), 50(2), p. 187-228 (2013). | MR
[64] Kowalski (E.).— Sieve in expansion. Séminaire Bourbaki, 63(1028), p. 1-35 (2011).
[65] Landau (E.).— Über die Klassenzahl imaginär-quadratischer Zahlkörper. Nachr. Ges. Wiss. Gottingen, p. 285-295 (1918).
[66] Landau (E.).— Bemerkungen zum Heilbronnschen Satz. Acta Arith, p. 1-18 (1935).
[67] Linnik (U. V.).— The large sieve. C. R. (Doklady) Acad. Sci. URSS (N.S.), 30, p. 292-294 (1941). | MR
[68] Lax (P.D.) and Phillips (R.S.).— The asymptotic distribution of lattice points in Euclidean and non-Euclidean space. Journal of Functional Analysis, 46, p. 280-350 (1982). | MR | Zbl
[69] Luo (W.), Rudnick (Z.), and Sarnak (P.).— On Selberg’s eigenvalue conjecture. Geom. Funct. Anal., 5(2), p. 387-401 (1995). | MR | Zbl
[70] Liu (J.) and Sarnak (P.).— Integral points on quadrics in three variables whose coordinates have few prime factors. Israel J. Math, 178, p. 393-426 (2010). | MR | Zbl
[71] Lubotzky (A.).— Expander graphs in pure and applied mathematics. Bull. Amer. Math. Soc., 49, p. 113-162 (2012). | MR | Zbl
[72] Maier (H.).— Primes in short intervals. Michigan Math. J., 32(2), p. 221-225 (1985). | MR | Zbl
[73] Maier (H.).— Small differences between prime numbers. Michigan Math J., 35, p. 323-344 (1988). | MR | Zbl
[74] Maynard (J.).— Small gaps between primes (2013). Preprint, arXiv:1311.4600.
[75] McMullen (C. T.).— Uniformly Diophantine numbers in a fixed real quadratic field. Compos. Math., 145(4), p. 827-844 (2009). | MR | Zbl
[76] McMullen (C. T.).— Dynamics of units and packing constants of ideals, 2012. Online lecture notes, http://www.math.harvard.edu/ ctm/ expositions/home/text/papers/cf/slides/slides.pdf.
[77] Montgomery (H. L.).— Topics in Multiplicative Number Theory, volume 227 of Lecture Notes in Math. Springer, New York (1971). | MR | Zbl
[78] Mordell (L. J.).— On the riemann hypothesis and imaginary quadratic fields with a given class number. J. London Math. Soc., 9, p. 289-298 (1934). | MR | Zbl
[79] Motohashi (Y) and Pintz (J.).— A smoothed GPY sieve. Bull. Lond. Math. Soc., 40(2), p. 298-310 (2008). | MR | Zbl
[80] Novikov (P. S.).— Ob algoritmiěskoǐ nerazrešimosti problemy toždestva slov v teorii grupp. Trudy Mat. Inst. im. Steklov. no. 44. Izdat. Akad. Nauk SSSR, Moscow (1955). | MR | Zbl
[81] Nevo (A.) and Sarnak (P.).— Prime and almost prime integral points on principal homogeneous spaces (2009). | MR | Zbl
[82] Pyber (L.) and Szabo (E.).— Growth in finite simple groups of lie type of bounded rank, 2010. Preprint arXiv:1005.1858.
[83] Rényi (A.).— On the representation of an even number as the sum of a single prime and single almost-prime number. Izvestiya Akad. Nauk SSSR. Ser. Mat., 12, p. 57-78 (1948). | MR | Zbl
[84] Rankin (R. A.).— The difference between consecutive prime numbers. II. Proc. Cambridge Philos. Soc., 36, p. 255-266 (1940). | MR
[85] Riemann (B.).— Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsberichte der Berliner Akademie (1859).
[86] Roth (K.F.).— On the large sieves of Linnik and Rényi. Mathematika, 12, p. 1-9 (1965). | MR | Zbl
[87] Sarnak (P.).— Selberg’s eigenvalue conjecture. Notices Amer. Math. Soc., 42(11), p. 1272-1277 (1995). | MR | Zbl
[88] Sarnak (P.).— What is... an expander? Notices Amer. Math. Soc., 51(7), p. 762-763 (2004). | MR | Zbl
[89] Sarnak (P.).— Notes on the generalized Ramanujan conjectures. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 659-685. Amer. Math. Soc., Providence, RI (2005). | MR | Zbl
[90] Sarnak (P.).— Letter to J. Lagarias (2007). http://web.math.princeton. edu/sarnak/AppolonianPackings.pdf. | MR
[91] Sarnak (P.).— Equidistribution and primes. Astérisque, (322), p. 225-240 (2008). Géométrie différentielle, physique mathématique, mathématiques et société. II. | MR | Zbl
[92] Sarnak (P.).— Affine sieve (2010). Slides from lectures, http://www.math. princeton.edu/sarnak/Affinesievesummer2010.pdf.
[93] Sarnak (P.).— Notes on thin matrix groups. In Thin Groups and Superstrong Approximation, volume 61 of Mathematical Sciences Research Institute Publications, p. 343-362. Cambridge University Press (2014). | MR
[94] Selberg (A.).— On the estimation of Fourier coefficients of modular forms. Proc. of Symposia in Pure Math., VII, p. 1-15 (1965). | MR | Zbl
[95] Serre (J.-P.).— Topics in Galois theory, volume 1 of Res. Notes in Math. A.K. Peters (2008). | MR | Zbl
[96] Salehi Golsefidy (A.).— Affine sieve and expanders (2012). To appear, Proceedings of MSRI.
[97] Salehi Golsefidy (A.) and Sarnak (P.).— Affine sieve (2011). To appear, JAMS.
[98] Salehi Golsefidy (A.) and Varjú (P. P.).— Expansion in perfect groups. Geom. Funct. Anal., 22(6), p. 1832-1891 (2012). | MR | Zbl
[99] Siegel (C. L.).— Über die Classenzahl quadratischer Zahlkörper. Acta Arith, 1, p. 83-86 (1935).
[100] Soundararajan (K.).— Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim. Bull. Amer. Math. Soc. (N.S.), 44(1), p. 1-18 (2007). | MR | Zbl
[101] Sarnak (P.) and Xue (X.).— Bounds for multiplicities of automorphic representations. Duke J. Math., 64(1), p. 207-227 (1991). | MR | Zbl
[102] Vinogradov (A. I.).— The density hypothesis for Dirichet L-series. Izv. Akad. Nauk SSSR Ser. Mat., 29, p. 903-934, 1965. | MR | Zbl
[103] Walfisz (A.).— Zur additiven Zahlentheorie. II. Math. Z., 40(1), p. 592-607 (1936). | MR
[104] Zhang (Y.).— Bounded gaps between primes (2013). To appear, Annals Math. 2 | MR | Zbl
Cited by Sources: