# Difference between revisions of "Bounded gaps between primes"

## World records

• $H$ is a quantity such that there are infinitely many pairs of consecutive primes of distance at most $H$ apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
• $k_0$ is a quantity such that every admissible $k_0$-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in $k_0$ lead to improvements in $H$. (The relationship is roughly of the form $H \sim k_0 \log k_0$.
• $\varpi$ is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in $\varpi$ lead to improvements in $k_0$. (The relationship is roughly of the form $k_0 \sim \varpi^{-3/2}$.)
Date $\varpi$ $k_0$ $H$ Comments
14 May 1/1,168 (Zhang) 3,500,000 (Zhang) 70,000,000 (Zhang) All subsequent work is based on Zhang's breakthrough paper.
21 May 63,374,611 (Lewko) Optimises Zhang's condition $\pi(H)-\pi(k_0) \gt k_0$; can be reduced by 1 by parity considerations
28 May 59,874,594 (Trudgian) Uses $(p_{m+1},\ldots,p_{m+k_0})$ with $p_{m+1} \gt k_0$
30 May 59,470,640 (Morrison)

58,885,998? (Tao)

59,093,364 (Morrison)

57,554,086 (Morrison)

Uses $(p_{m+1},\ldots,p_{m+k_0})$ and then $(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})$ following [HR1973], [HR1973b], [R1974] and optimises in m
31 May 2,947,442 (Morrison)

2,618,607 (Morrison)

48,112,378 (Morrison)

42,543,038 (Morrison)

42,342,946 (Morrison)

Optimizes Zhang's condition $\omega\gt0$, and then uses an improved bound on $\delta_2$
1 Jun 42,342,924 (Tao) Tiny improvement using the parity of $k_0$
2 Jun 866,605 (Morrison) 13,008,612 (Morrison) Uses a further improvement on the quantity $\Sigma_2$ in Zhang's analysis (replacing the previous bounds on $\delta_2$)
3 Jun 1/1,040? (v08ltu) 341,640 (Morrison) 4,982,086 (Morrison)

4,802,222 (Morrison)

Uses a different method to establish $DHL[k_0,2]$ that removes most of the inefficiency from Zhang's method.
4 Jun 1/224?? (v08ltu)

1/240?? (v08ltu)

4,801,744 (Sutherland)

4,788,240 (Sutherland)

Uses asymmetric version of the Hensley-Richards tuples
5 Jun 34,429? (Paldi/v08ltu)

34,429 (Tao/v08ltu/Harcos)

4,725,021 (Elsholtz)

4,717,560 (Sutherland)

397,110? (Sutherland)

4,656,298 (Sutherland)

389,922 (Sutherland)

388,310 (Sutherland)

388,284 (Castryck)

388,248 (Sutherland)

387,982 (Castryck)

387,974 (Castryck)

k_0 bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.

H bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

6 Jun 387,960 (Angelveit)

Experimentation with different residue classes and different intervals

? - unconfirmed or conditional

?? - theoretical limit of an analysis, rather than a claimed record

## Benchmarks

Let H be the minimal diameter of an admissible tuple of cardinality $k_0 = 34,429$. For benchmark upper bounds:

• The Zhang sieve (as optimized by Trudgian and later Morrison) gives $H \leq 411,932$.
• The Hensley-Richards sieve gives $H \leq 402,790$ after optimization.
• The shifted Hensley-Richards sieve gives $H \leq 401,700$ after optimization.

For benchmark lower bounds:

• The easier version of the Montgomery-Vaughan large sieve inequality $k_0 \sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H+1+Q^2$ gives $H \geq 196,729$.
• The Montgomery-Vaughan Brun-Titchmarsh bound $k_0 \leq \frac{2(H+1)}{\log(H+1)}$ gives $H \geq 211,046$.
• The Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives $H \geq 226,987$ after optimization.
• Using a refinement of this inequality [B1995, p.162] improves this to $H \geq 227,078$.

## Bibliography

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

• [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. MathSciNet
• [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. MathSciNet Article
• [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. MathSciNet Article
• [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
• [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 MathSciNet Article
• [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). MathSciNet Article
• [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. MathSciNet Article
• [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. JSTOR
• [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. arXiv MathSciNet
• [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. MathSciNet Article
• [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. MathSciNet Article
• [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. MathSciNet Article
• [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. arXiv MathSciNet
• [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. MathSciNet Article
• [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. MathSciNet Article
• [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. MathSciNet Article
• [S2007] 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. MathSciNet Article arXiv

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