Primes in intervals of bounded length

Author:
Andrew Granville

Journal:
Bull. Amer. Math. Soc. **52** (2015), 171-222

MSC (2010):
Primary 11P32

Published electronically:
February 11, 2015

MathSciNet review:
3312631

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Abstract | References | Similar Articles | Additional Information

Abstract: The infamous *twin prime conjecture* states that there are infinitely many pairs of distinct primes which differ by . Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound such that there are infinitely many pairs of distinct primes which differ by no more than . This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable.

Zhang even showed that one can take . Moreover, a co-operative team, *Polymath8*, collaborating only online, had been able to lower the value of to . They had not only been more careful in several difficult arguments in Zhang's original paper, they had also developed Zhang's techniques to be both more powerful and to allow a much simpler proof (and this forms the basis for the proof presented herein).

In November 2013, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to , by a substantially easier method. Both Maynard and Terry Tao, who had independently developed the same idea, were able to extend their proofs to show that for any given integer there exists a bound such that there are infinitely many intervals of length containing at least distinct primes. We will also prove this much stronger result herein, even showing that one can take .

If Zhang's method is combined with the Maynard-Tao setup, then it appears that the bound can be further reduced to . If all of these techniques could be pushed to their limit, then we would obtain () (or arguably to ), so new ideas are still needed to have a feasible plan for proving the twin prime conjecture.

The article will be split into two parts. The first half will introduce the work of Zhang, Polymath8, Maynard and Tao, and explain their arguments that allow them to prove their spectacular results. The second half of this article develops a proof of Zhang's main novel contribution, an estimate for primes in relatively short arithmetic progressions.

- [1]
W. D. Banks, T. Freiberg, J. Maynard,
*On limit points of the sequence of normalized prime gaps*, preprint. - [2]
W. D. Banks, T. Freiberg, C. L. Turnage-Butterbaugh,
*Consecutive primes in tuples*, preprint. **[3]**E. Bombieri,*On the large sieve*, Mathematika**12**(1965), 201–225. MR**0197425****[4]**E. Bombieri and H. Davenport,*Small differences between prime numbers*, Proc. Roy. Soc. Ser. A**293**(1966), 1–18. MR**0199165****[5]**E. Bombieri, J. B. Friedlander, and H. Iwaniec,*Primes in arithmetic progressions to large moduli*, Acta Math.**156**(1986), no. 3-4, 203–251. MR**834613**, 10.1007/BF02399204**[6]**E. Bombieri, J. B. Friedlander, and H. Iwaniec,*Primes in arithmetic progressions to large moduli. II*, Math. Ann.**277**(1987), no. 3, 361–393. MR**891581**, 10.1007/BF01458321**[7]**E. Bombieri, J. B. Friedlander, and H. Iwaniec,*Primes in arithmetic progressions to large moduli. III*, J. Amer. Math. Soc.**2**(1989), no. 2, 215–224. MR**976723**, 10.1090/S0894-0347-1989-0976723-6- [8]
A. Castillo, C. Hall, R. J. Lemke Oliver, P. Pollack, L. Thompson,
*Bounded gaps between primes in number fields and function fields*, preprint. - [9]
L. Chua, S. Park, G. D. Smith,
*Bounded gaps between primes in special sequences*, preprint. **[10]**Pierre Deligne,*La conjecture de Weil. II*, Inst. Hautes Études Sci. Publ. Math.**52**(1980), 137–252 (French). MR**601520****[11]**J.-M. Deshouillers and H. Iwaniec,*Kloosterman sums and Fourier coefficients of cusp forms*, Invent. Math.**70**(1982/83), no. 2, 219–288. MR**684172**, 10.1007/BF01390728**[12]**P. D. T. A. Elliott and H. Halberstam,*A conjecture in prime number theory*, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 59–72. MR**0276195**- [13]
K. Ford, B. Green, S. Konyagin and T. Tao,
*Large gaps between consecutive prime numbers*, preprint. - [14]
E. Fouvry,
*A new form of the error term in the linear sieve*, Acta Arith.,**37**(1980), 307-320. **[15]**Étienne Fouvry,*Autour du théorème de Bombieri-Vinogradov*, Acta Math.**152**(1984), no. 3-4, 219–244 (French). MR**741055**, 10.1007/BF02392198**[16]**E. Fouvry and H. Iwaniec,*On a theorem of Bombieri-Vinogradov type*, Mathematika**27**(1980), no. 2, 135–152 (1981). MR**610700**, 10.1112/S0025579300010032**[17]**E. Fouvry and H. Iwaniec,*Primes in arithmetic progressions*, Acta Arith.**42**(1983), no. 2, 197–218. MR**719249**- [18]
T. Freiberg,
*A note on the theorem of Maynard and Tao*, preprint. **[19]**John Friedlander and Andrew Granville,*Limitations to the equi-distribution of primes. I*, Ann. of Math. (2)**129**(1989), no. 2, 363–382. MR**986796**, 10.2307/1971450**[20]**John B. Friedlander and Henryk Iwaniec,*Incomplete Kloosterman sums and a divisor problem*, Ann. of Math. (2)**121**(1985), no. 2, 319–350. With an appendix by Bryan J. Birch and Enrico Bombieri. MR**786351**, 10.2307/1971175- [21]
J. Friedlander and H. Iwaniec,
*Close encounters among the primes*, preprint. **[22]**P. X. Gallagher,*Bombieri’s mean value theorem*, Mathematika**15**(1968), 1–6. MR**0237442****[23]**Daniel A. Goldston, János Pintz, and Cem Y. Yıldırım,*Primes in tuples. I*, Ann. of Math. (2)**170**(2009), no. 2, 819–862. MR**2552109**, 10.4007/annals.2009.170.819**[24]**D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim,*Small gaps between primes or almost primes*, Trans. Amer. Math. Soc.**361**(2009), no. 10, 5285–5330. MR**2515812**, 10.1090/S0002-9947-09-04788-6**[25]**Solomon Wolf Golomb,*PROBLEMS IN THE DISTRIBUTION OF THE PRIME NUMBERS*, ProQuest LLC, Ann Arbor, MI, 1957. Thesis (Ph.D.)–Harvard University. MR**2938872****[26]**S. W. Graham and C. J. Ringrose,*Lower bounds for least quadratic nonresidues*, Analytic number theory (Allerton Park, IL, 1989) Progr. Math., vol. 85, Birkhäuser Boston, Boston, MA, 1990, pp. 269–309. MR**1084186**- [27]
A. Granville, D.M. Kane, D. Koukoulopoulos, and R. Lemke Oliver,
*Best possible densities, as a consequence of Zhang-Maynard-Tao*, to appear. - [28]
A. Granville and K. Soundararajan,
*Multiplicative number theory; the alternative approach*, to appear. **[29]**Ben Green and Terence Tao,*The primes contain arbitrarily long arithmetic progressions*, Ann. of Math. (2)**167**(2008), no. 2, 481–547. MR**2415379**, 10.4007/annals.2008.167.481**[30]**Ben Green, Terence Tao, and Tamar Ziegler,*An inverse theorem for the Gowers 𝑈^{𝑠+1}[𝑁]-norm*, Ann. of Math. (2)**176**(2012), no. 2, 1231–1372. MR**2950773**, 10.4007/annals.2012.176.2.11**[31]**Rajiv Gupta and M. Ram Murty,*A remark on Artin’s conjecture*, Invent. Math.**78**(1984), no. 1, 127–130. MR**762358**, 10.1007/BF01388719**[32]**G. H. Hardy and J. E. Littlewood,*Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes*, Acta Math.**44**(1923), no. 1, 1–70. MR**1555183**, 10.1007/BF02403921**[33]**D. R. Heath-Brown,*Prime numbers in short intervals and a generalized Vaughan identity*, Canad. J. Math.**34**(1982), no. 6, 1365–1377. MR**678676**, 10.4153/CJM-1982-095-9**[34]**D. R. Heath-Brown,*Artin’s conjecture for primitive roots*, Quart. J. Math. Oxford Ser. (2)**37**(1986), no. 145, 27–38. MR**830627**, 10.1093/qmath/37.1.27**[35]**D. R. Heath-Brown,*The largest prime factor of 𝑋³+2*, Proc. London Math. Soc. (3)**82**(2001), no. 3, 554–596. MR**1816689**, 10.1112/plms/82.3.554- [36]
H. A. Helfgott,
*Major arcs for Goldbach's theorem*, to appear. **[37]**Douglas Hensley and Ian Richards,*On the incompatibility of two conjectures concerning primes*, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 123–127. MR**0340194****[38]**Douglas Hensley and Ian Richards,*Primes in intervals*, Acta Arith.**25**(1973/74), 375–391. MR**0396440****[39]**Christopher Hooley,*On Artin’s conjecture*, J. Reine Angew. Math.**225**(1967), 209–220. MR**0207630****[40]**Henryk Iwaniec and Emmanuel Kowalski,*Analytic number theory*, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR**2061214**- [41]
H. D. Kloosterman,
*On the representation of numbers in the form*, Acta Mathematica**49**(1926), pp. 407-464. - [42]
E. Kowalski,
*Gaps between prime numbers and primes in arithmetic progressions (after Y. Zhang and J. Maynard)*, Séminaire Bourbaki**66**(2013-2014), no. 1084. - [43]
H. Li, H. Pan,
*Bounded gaps between primes of the special form*, preprint. **[44]**Helmut Maier,*Small differences between prime numbers*, Michigan Math. J.**35**(1988), no. 3, 323–344. MR**978303**, 10.1307/mmj/1029003814- [45]
J. Maynard,
*Small gaps between primes*, to appear, Annals Math. - [46]
J. Maynard,
*Large gaps between primes*, preprint. - [47]
J. Maynard,
*Dense clusters of primes in subsets*, preprint. - [48]
L. J. Mordell,
*On a sum analogous to a Gauss's sum*, Quart. J. Math. Oxford Ser.**3**(1932), 161-167. **[49]**Yoichi Motohashi,*An induction principle for the generalization of Bombieri’s prime number theorem*, Proc. Japan Acad.**52**(1976), no. 6, 273–275. MR**0422179****[50]**Yoichi Motohashi and János Pintz,*A smoothed GPY sieve*, Bull. Lond. Math. Soc.**40**(2008), no. 2, 298–310. MR**2414788**, 10.1112/blms/bdn023- [51]
J. Pintz,
*Polignac Numbers, Conjectures of Erdős on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture*, preprint. - [52]
J. Pintz,
*On the ratio of consecutive gaps between primes*, preprint. - [53]
J. Pintz,
*On the distribution of gaps between consecutive primes*, preprint. - [54]
A. De Polignac,
*Six propositions arithmologiques déduites du crible d'Ératosothène*, Nouvelles annales de mathématiques**8**(1849), 423-429. - [55]
P. Pollack,
*Bounded gaps between primes with a given primitive root*, preprint. - [56]
P. Pollack and L. Thompson,
*Arithmetic functions at consecutive shifted primes*, preprint. - [57]
D. H. J. Polymath,
*New equidistribution estimates of Zhang type*, preprint. - [58]
D. H. J. Polymath,
*Variants of the Selberg sieve, and bounded intervals containing many primes*, preprint. **[59]**A. Schinzel,*Remarks on the paper “Sur certaines hypothèses concernant les nombres premiers”*, Acta Arith.**7**(1961/1962), 1–8. MR**0130203**- [60]
A. Selberg,
*On elementary methods in prime number theory and their limitations*, in Proc. 11th Scand. Math. Cong. Trondheim (1949), Collected Works, Vol. I, 388-477, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1989. **[61]**P. Shiu,*A Brun-Titchmarsh theorem for multiplicative functions*, J. Reine Angew. Math.**313**(1980), 161–170. MR**552470**, 10.1515/crll.1980.313.161**[62]**K. Soundararajan,*Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım*, Bull. Amer. Math. Soc. (N.S.)**44**(2007), no. 1, 1–18. MR**2265008**, 10.1090/S0273-0979-06-01142-6- [63]
T. Tao,
*private communication*. - [64]
J. Thorner,
*Bounded gaps between primes in Chebotarev sets*, preprint. **[65]**Robert-C. Vaughan,*Sommes trigonométriques sur les nombres premiers*, C. R. Acad. Sci. Paris Sér. A-B**285**(1977), no. 16, A981–A983 (French, with English summary). MR**0498434****[66]**A. I. Vinogradov,*The density hypothesis for Dirichet 𝐿-series*, Izv. Akad. Nauk SSSR Ser. Mat.**29**(1965), 903–934 (Russian). MR**0197414****[67]**André Weil,*Numbers of solutions of equations in finite fields*, Bull. Amer. Math. Soc.**55**(1949), 497–508. MR**0029393**, 10.1090/S0002-9904-1949-09219-4**[68]**Yitang Zhang,*Bounded gaps between primes*, Ann. of Math. (2)**179**(2014), no. 3, 1121–1174. MR**3171761**, 10.4007/annals.2014.179.3.7

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Additional Information

**Andrew Granville**

Affiliation:
Département de mathématiques et de statistiques, Université de Montréal, Montréal QC H3C 3J7, Canada

Email:
andrew@dms.umontreal.ca

DOI:
http://dx.doi.org/10.1090/S0273-0979-2015-01480-1

Received by editor(s):
September 5, 2014

Published electronically:
February 11, 2015

Additional Notes:
This article was shortened for final publication and several important references, namely [7, 10, 14, 16, 37, 38, 41, 43, 48, 53, 59, 60, 61, 65, 66, 67], are no longer directly referred to in the text. Nonetheless we leave these references here for the enthusiastic student.

Dedicated:
To Yitang Zhang, for showing that one can, no matter what

Article copyright:
© Copyright 2015
American Mathematical Society