# Timeline of prime gap bounds

(Difference between revisions)
 Revision as of 03:38, 18 February 2014 (view source)← Older edit Revision as of 03:39, 18 February 2014 (view source)Newer edit → Line 827: Line 827: | Jan 6 | Jan 6 | | - | 54? ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen]) + | 54#? ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen]) - | 270? ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis]) + | 270#? ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis]) | | |- |-

## Revision as of 03:39, 18 February 2014

Date$\varpi$ or $(\varpi,\delta)$ k0 H Comments
Aug 10 2005 6 [EH] 16 [EH] ([Goldston-Pintz-Yildirim]) First bounded prime gap result (conditional on Elliott-Halberstam)
May 14 2013 1/1,168 (Zhang) 3,500,000 (Zhang) 70,000,000 (Zhang) All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
May 21 63,374,611 (Lewko) Optimises Zhang's condition π(H) − π(k0) > k0; can be reduced by 1 by parity considerations
May 28 59,874,594 (Trudgian) Uses $(p_{m+1},\ldots,p_{m+k_0})$ with pm + 1 > k0
May 30 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
May 31 2,947,442 (Morrison)

2,618,607 (Morrison)

48,112,378 (Morrison)

42,543,038 (Morrison)

42,342,946 (Morrison)

Optimizes Zhang's condition ω > 0, and then uses an improved bound on δ2
Jun 1 42,342,924 (Tao) Tiny improvement using the parity of k0
Jun 2 866,605 (Morrison) 13,008,612 (Morrison) Uses a further improvement on the quantity Σ2 in Zhang's analysis (replacing the previous bounds on δ2)
Jun 3 1/1,040? (v08ltu) 341,640 (Morrison) 4,982,086 (Morrison)

4,802,222 (Morrison)

Uses a different method to establish DHL[k0,2] that removes most of the inefficiency from Zhang's method.
Jun 4 1/224?? (v08ltu)

1/240?? (v08ltu)

4,801,744 (Sutherland)

4,788,240 (Sutherland)

Uses asymmetric version of the Hensley-Richards tuples
Jun 5 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)

k0 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

Jun 6 (1/488,3/9272) (Pintz)

1/552 (Pintz, Tao)

60,000* (Pintz)

52,295* (Peake)

11,123 (Tao)

387,960 (Angelveit)

387,904 (Angeltveit)

768,534* (Pintz)

Improved H-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
Jun 7 (1/538, 1/660) (v08ltu)

(1/538, 31/20444) (v08ltu)

(1/942, 19/27004) (v08ltu)

$828 \varpi + 172\delta < 1$ (v08ltu/Green)

11,018 (Tao)

10,721 (v08ltu)

10,719 (v08ltu)

25,111 (v08ltu)

26,024? (vo8ltu)

113,520? (Angeltveit)

116,386* (Sun)

275,262 (Castryck-pedant-Sutherland)

275,388* (xfxie-Sutherland)

275,126 (Castryck-pedant-Sutherland)

274,970 (Castryck-pedant-Sutherland)

275,208* (xfxie)

387,534 (pedant-Sutherland)

Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
Jun 8 286,224 (Sutherland)

285,752 (pedant-Sutherland)

values of $\varpi,\delta,k_0$ now confirmed; most tuples available on dropbox. New bounds on H obtained via iterated merging using a randomized greedy sieve.
Jun 9 181,000*? (Pintz) 2,530,338*? (Pintz) New bounds on H obtained by interleaving iterated merging with local optimizations.
Jun 10 23,283? (Harcos/v08ltu) 285,210 (Sutherland) More efficient control of the κ error using the fact that numbers with no small prime factor are usually coprime
Jun 11 252,804 (Sutherland) More refined local "adjustment" optimizations, as detailed here.

An issue with the k0 computation has been discovered, but is in the process of being repaired.

Jun 12 22,951 (Tao/v08ltu)

22,949 (Harcos)

249,180 (Castryck) Improved bound on k0 avoids the technical issue in previous computations.
Jun 13
Jun 14 248,898 (Sutherland)
Jun 15 $348\varpi+68\delta < 1$? (Tao) 6,330? (v08ltu)

6,329? (Harcos)

6,329 (v08ltu)

60,830? (Sutherland) Taking more advantage of the α convolution in the Type III sums
Jun 16 $348\varpi+68\delta < 1$ (v08ltu)

155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 (Tao)

3,405 (v08ltu) 60,760* (Sutherland)

30,606 (Engelsma)

Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
Jun 18 5,937? (Pintz/Tao/v08ltu)

5,672? (v08ltu)

5,459? (v08ltu)

5,454? (v08ltu)

5,453? (v08ltu)

60,740 (xfxie)

60,726 (xfxie-Sutherland)

58,866? (Sun)

53,898? (Sun)

53,842? (Sun)

A new truncated sieve of Pintz virtually eliminates the influence of δ
Jun 19 5,455? (v08ltu)

5,453? (v08ltu)

5,452? (v08ltu)

53,774? (Sun)

53,672*? (Sun)

Some typos in κ3 estimation had placed the 5,454 and 5,453 values of k0 into doubt; however other refinements have counteracted this
Jun 20 $178\varpi + 52\delta < 1$? (Tao)

$148\varpi + 33\delta < 1$? (Tao)

Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
Jun 21 $148\varpi + 33\delta < 1$ (v08ltu) 1,470 (v08ltu)

1,467 (v08ltu)

12,042 (Engelsma) Systematic tables of tuples of small length have been set up here and here (update: As of June 27 these tables have been merged and uploaded to an online database of current bounds on H(k) for k up to 5000).
Jun 22 1,466 (Harcos/v08ltu) 12,006 (Engelsma) Slight improvement in the $\tilde \theta$ parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of k0
Jun 23 1,466 (Paldi/Harcos) 12,006 (Engelsma) An improved monotonicity formula for $G_{k_0-1,\tilde \theta}$ reduces κ3 somewhat
Jun 24 $(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1$? (v08ltu)

$140\varpi + 32 \delta < 1$? (Tao)

1/88?? (Tao)

1/74?? (Tao)

1,268? (v08ltu) 10,206? (Engelsma) A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
Jun 25 $116\varpi+30\delta<1$? (Fouvry-Kowalski-Michel-Nelson/Tao) 1,346? (Hannes)

502?? (Trevino)

1,007? (Hannes)

10,876? (Engelsma)

3,612?? (Engelsma)

7,860? (Engelsma)

Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
Jun 26 $116\varpi + 25.5 \delta < 1$? (Nielsen)

$(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1$? (Tao)

962? (Hannes) 7,470? (Engelsma) Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see this page, in particular optimising van der Corput in the Type I sums. Integrated tuples page now online.
Jun 27 $108\varpi + 30 \delta < 1$? (Tao) 902? (Hannes) 6,966? (Engelsma) Improved the Type III estimates by averaging in α; also some slight improvements to the Type II sums. Tuples page is now accepting submissions.
Jul 1 $(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1$? (Tao)

873? (Hannes)

872? (xfxie)

Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
Jul 5 $(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1$ (Tao)

720 (xfxie/Harcos)

Weakened the assumption of xδ-smoothness of the original moduli to that of double xδ-dense divisibility

Jul 10 7/600? (Tao) An in principle refinement of the van der Corput estimate based on exploiting additional averaging
Jul 19 $(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1$? (Tao) A more detailed computation of the Jul 10 refinement
Jul 20 Jul 5 computations now confirmed
Jul 27 633 (Tao)

632 (Harcos)

4,686 (Engelsma)
Jul 30 $168\varpi + 48\delta < 1$# (Tao) 1,788# (Tao) 14,994# (Sutherland) Bound obtained without using Deligne's theorems.
Aug 17 1,783# (xfxie) 14,950# (Sutherland)
Oct 3 13/1080?? (Nelson/Michel/Tao) 604?? (Tao) 4,428?? (Engelsma) Found an additional variable to apply van der Corput to
Oct 11 $83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1$? (Tao) 603? (xfxie) 4,422?(Engelsma)

12 [EH] (Maynard)

Worked out the dependence on δ in the Oct 3 calculation
Oct 21 All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
Oct 23 700#? (Maynard) Announced at a talk in Oberwolfach
Oct 24 110#? (Maynard) 628#? (Clark-Jarvis) With this value of k0, the value of H given is best possible (and similarly for smaller values of k0)
Nov 19 105# (Maynard)

5 [EH] (Maynard)

600# (Maynard/Clark-Jarvis) One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
Nov 20 145*? (Nielsen)

13,986 [m=2]#? (Nielsen)

864*? (Clark-Jarvis)

145,212 [m=2]#? (Sutherland)

Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
Nov 21 68?? (Maynard)

582#*? (Nielsen])

59,451 [m=2]#? (Nielsen])

508*? (xfxie)

42,392 [m=2]? (Nielsen)

356?? (Clark-Jarvis) Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
Nov 22 388*? (xfxie)

448#*? (Nielsen)

43,134 [m=2]#? (Nielsen)

698,288 [m=2]#? (Sutherland)

484,290 [m=2]? (Sutherland)

484,276 [m=2]? (Sutherland)

Uses the m=2 values of k_0 from Nov 21
Nov 23 493,528 [m=2]#? Sutherland

493,510 [m=2]#? Sutherland

484,272 [m=2]? (xfxie)

484,260 [m=2]? (Sutherland)

484,238 [m=2]? (xfxie)

493,458 [m=2]#? Sutherland

Nov 24 484,234 [m=2]? (Sutherland)

484,200 [m=2]? (xfxie)

493,442 [m=2]#? (Sutherland)

484,192 [m=2]? (Sutherland)

Nov 25 385#*? (xfxie)

339*? (xfxie)

484,176 [m=2]? (Sutherland)

493,436[m=2]#? (Sutherland)

Using the exponential moment method to control errors
Nov 26 102# (Nielsen) 493,426 [m=2]#? (Sutherland)

484,168 [m=2]? (xfxie)

576# (Clark-Jarvis)

Optimising the original Maynard variational problem
Nov 27 484,162 [m=2]? (Sutherland)

484,142 [m=2]? (Sutherland)

Nov 28 484,136 [m=2]? (Sutherland

484,126 [m=2]? (Sutherland)

Dec 4 64#? (Nielsen) 330#? (Clark-Jarvis) Searching over a wider range of polynomials than in Maynard's paper
Dec 6 493,408 [m=2]#? (Sutherland)
Dec 19 59#? (Nielsen)

10,000,000? [m=3] (Tao)

1,700,000? [m=3] (Tao)

38,000? [m=2] (Tao)

300#? (Clark-Jarvis)

182,087,080? [m=3] (Sutherland)

179,933,380? [m=3] (Sutherland)

More efficient memory management allows for an increase in the degree of the polynomials used; the m=2,3 results use an explicit version of the $M_k \geq \frac{k}{k-1} \log k - O(1)$ lower bound.
Dec 20 25,819? [m=2] (Castryck)

55#? (Nielsen)

36,000? [m=2] (xfxie)

35,146? [m=2] (xfxie)

175,225,874? [m=3] (Sutherland)

27,398,976? [m=3] (Sutherland)

26,682,014? [m=3] (Sutherland)

431,682? [m=2] (Sutherland)

430,448? [m=2] (Sutherland)

429,822? [m=2] (Sutherland)

283,242? [m=2] (Sutherland)

272#? (Clark-Jarvis)

Dec 21 1,640,042? [m=3] (Sutherland)

41,862,295? [m=4] (Sutherland)

1,631,027? [m=3] (Sutherland)

1,630,680? [m=3] (xfxie)

36,000,000? [m=4] (xfxie

35,127,242? [m=4] (Sutherland)

25,589,558? [m=4] (xfxie)

429,798? [m=2] (Sutherland)

25,602,438? [m=3] (Sutherland)

405,528? [m=2] (Sutherland)

825,018,354? [m=4] (Sutherland)

25,533,684? [m=3] (Sutherland)

395,264? [m=2] (Sutherland)

395,234? [m=2] (xfxie)

395,178? [m=2] (Sutherland)

25,527,718? [m=3] (Sutherland)

685,833,596? [m=4] (Sutherland)

491,149,914? [m=4] (Sutherland)

24,490,758? [m=3] (Sutherland)

Optimising the explicit lower bound $M_k \geq \log k-O(1)$
Dec 22 1,628,944? [m=3] (Castryck)

75,000,000? [m=4] (Castryck)

3,400,000,000? [m=5] (Castryck)

5,511? [EH] [m=3] (Sutherland)

2,114,964#? [m=3] (Sutherland)

309,954? [EH] [m=5] (Sutherland)

74,487,363? [m=4] (xfxie)

1,628,943? [m=3] (xfxie)

395,154? [m=2] (Sutherland)

24,490,410? [m=3] (Sutherland)

485,825,850? [m=4] (Sutherland)

395,122? [m=2] (Sutherland)

473,244,502? [m=4] (Sutherland)

1,523,781,850? [m=4] (Sutherland)

82,575,303,678? [m=5] (Sutherland)

52,130? [EH] [m=3] (Sutherland)

33,661,442?# [m=3] (Sutherland)

24,462,790? [m=3] (Sutherland)

4,316,446? [EH] [m=5] (Sutherland)

A numerical precision issue was discovered in the earlier m=4 calculations
Dec 23 41,589? [EH] [m=4] (Sutherland)

41,588? [EH] [m=4] (xfxie)

309,661? [EH] [m=5] (xfxie)

105,754,838#? [m=4] (Sutherland)

5,300,000#? [m=5] (Sutherland)

24,462,774? [m=3] (Sutherland)

1,512,832,950? [m=4] (Sutherland)

4,146,936? [EH] [m=5] (Sutherland)

52,116? [EH] [m=3] (Sutherland)

474,600? [EH] [m=4] (Sutherland)

474,460? [EH] [m=4] (Sutherland)

4,143,140? [EH] [m=5] (Sutherland)

32,313,942#? [m=3] (Sutherland)

2,186,561,568#? [m=4] (Sutherland)

474,372? [EH] [m=4] (Sutherland)

131,161,149,090#? [m=5] (Sutherland)

Dec 24 474,320? [EH] [m=4] (Sutherland)

4,137,872? [EH] [m=5] (Sutherland)

24,462,654? [m=3] (Sutherland)

1,497,901,734? [m=4] (Sutherland)

32,313,878#? [m=3] (Sutherland)

Dec 28 474,296? [EH] [m=4] (Sutherland)

4,137,854? [EH] [m=5] (Sutherland)

Jan 2 2014 474,290? [EH] [m=4] (Sutherland)
Jan 6 54#? (Nielsen) 270#? (Clark-Jarvis)
Jan 8 4 [GEH] (Nielsen) 8 [GEH] (Nielsen) Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed here.
Jan 9 474,266? [EH] [m=4] (Sutherland)
Jan 28 395,106? [m=2] (Sutherland)
Jan 29 3 [GEH] (Nielsen) 6 [GEH] (Nielsen) A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube
Feb 9 Jan 29 results confirmed here
Feb 17 53?# (Nielsen) 264?# (Clark-Jarvis) Managed to get the epsilon trick to be computationally feasible for medium k

Legend:

1.  ? - unconfirmed or conditional
2.  ?? - theoretical limit of an analysis, rather than a claimed record
3. * - is majorized by an earlier but independent result
4. # - bound does not rely on Deligne's theorems
5. [EH] - bound is conditional the Elliott-Halberstam conjecture
6. [GEH] - bound is conditional the generalized Elliott-Halberstam conjecture
7. [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
8. strikethrough - values relied on a computation that has now been retracted

See also the article on Finding narrow admissible tuples for benchmark values of H for various key values of k0.