Difference between revisions of "Timeline of prime gap bounds"
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2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258451 Sutherland])  2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258451 Sutherland])  
+  
+  309,954? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258457 Sutherland])  
 [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258305 Sutherland])   [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258305 Sutherland])  
Line 758:  Line 760:  
[http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258452 Sutherland])  [http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258452 Sutherland])  
+  
+  [http://math.mit.edu/~drew/admissible_309954_4316446.txt 4,316,446]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8bivenlargingthesievesupportmoreefficientnumericsandexplicitasymptotics/#comment258457 Sutherland])  
 A numerical precision issue was discovered in the earlier m=4 calculations   A numerical precision issue was discovered in the earlier m=4 calculations  
}  } 
Revision as of 20:27, 22 December 2013
 [math]H = H_1[/math] is a quantity such that there are infinitely many pairs of consecutive primes of distance at most [math]H[/math] apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
 [math]k_0[/math] is a quantity such that every admissible [math]k_0[/math]tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in [math]k_0[/math] lead to improvements in [math]H[/math]. (The relationship is roughly of the form [math]H \sim k_0 \log k_0[/math]; see the page on finding narrow admissible tuples.) More recent improvements on [math]k_0[/math] have come from solving a Selberg sieve variational problem.
 [math]\varpi[/math] is a technical parameter related to a specialized form of the ElliottHalberstam conjecture. Would like to be as large as possible. Improvements in [math]\varpi[/math] lead to improvements in [math]k_0[/math], as described in the page on DicksonHardyLittlewood theorems. In more recent work, the single parameter [math]\varpi[/math] is replaced by a pair [math](\varpi,\delta)[/math] (in previous work we had [math]\delta=\varpi[/math]). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on distribution of primes in smooth moduli.
In this table, infinitesimal losses in [math]\delta,\varpi[/math] are ignored.
Date  [math]\varpi[/math] or [math](\varpi,\delta)[/math]  [math]k_0[/math]  [math]H[/math]  Comments 

10 Aug 2005  6 [EH]  16 [EH] ([GoldstonPintzYildirim])  First bounded prime gap result (conditional on ElliottHalberstam)  
14 May 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. 
21 May  63,374,611 (Lewko)  Optimises Zhang's condition [math]\pi(H)\pi(k_0) \gt k_0[/math]; can be reduced by 1 by parity considerations  
28 May  59,874,594 (Trudgian)  Uses [math](p_{m+1},\ldots,p_{m+k_0})[/math] with [math]p_{m+1} \gt k_0[/math]  
30 May  59,470,640 (Morrison)
58,885,998? (Tao) 59,093,364 (Morrison) 57,554,086 (Morrison) 
Uses [math](p_{m+1},\ldots,p_{m+k_0})[/math] and then [math](\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/21})[/math] 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 [math]\omega\gt0[/math], and then uses an improved bound on [math]\delta_2[/math]  
1 Jun  42,342,924 (Tao)  Tiny improvement using the parity of [math]k_0[/math]  
2 Jun  866,605 (Morrison)  13,008,612 (Morrison)  Uses a further improvement on the quantity [math]\Sigma_2[/math] in Zhang's analysis (replacing the previous bounds on [math]\delta_2[/math])  
3 Jun  1/1,040? (v08ltu)  341,640 (Morrison)  4,982,086 (Morrison)
4,802,222 (Morrison) 
Uses a different method to establish [math]DHL[k_0,2][/math] 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 HensleyRichards tuples  
5 Jun  34,429? (Paldi/v08ltu)  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) 
[math]k_0[/math] 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.
[math]H[/math] bound obtained by a hybrid Schinzel/greedy (or "greedygreedy") sieve  
6 Jun  

387,960 (Angelveit)
387,904 (Angeltveit)

Improved [math]H[/math]bounds based on experimentation with different residue classes and different intervals, and randomized tiebreaking in the greedy sieve. 
7 Jun 

26,024? (vo8ltu) 
387,534 (pedantSutherland) 
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 (pedantSutherland) 
values of [math]\varpi,\delta,k_0[/math] now confirmed; most tuples available on dropbox. New bounds on [math]H[/math] obtained via iterated merging using a randomized greedy sieve.  
Jun 9  181,000*? (Pintz)  2,530,338*? (Pintz)  New bounds on [math]H[/math] obtained by interleaving iterated merging with local optimizations.  
Jun 10  23,283? (Harcos/v08ltu)  285,210 (Sutherland)  More efficient control of the [math]\kappa[/math] 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 [math]k_0[/math] 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 [math]k_0[/math] avoids the technical issue in previous computations.  
Jun 13  
Jun 14  248,898 (Sutherland)  
Jun 15  [math]348\varpi+68\delta \lt 1[/math]? (Tao)  6,330? (v08ltu)
6,329? (Harcos) 6,329 (v08ltu) 
60,830? (Sutherland)  Taking more advantage of the [math]\alpha[/math] convolution in the Type III sums 
Jun 16  [math]348\varpi+68\delta \lt 1[/math] (v08ltu)


60,760* (Sutherland)

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)
58,866? (Sun) 53,898? (Sun) 53,842? (Sun) 
A new truncated sieve of Pintz virtually eliminates the influence of [math]\delta[/math]  
Jun 19  5,455? (v08ltu)
5,453? (v08ltu) 5,452? (v08ltu) 
53,774? (Sun)
53,672*? (Sun) 
Some typos in [math]\kappa_3[/math] estimation had placed the 5,454 and 5,453 values of [math]k_0[/math] into doubt; however other refinements have counteracted this  
Jun 20  [math]178\varpi + 52\delta \lt 1[/math]? (Tao)
[math]148\varpi + 33\delta \lt 1[/math]? (Tao) 
Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloostermantype sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli  
Jun 21  [math]148\varpi + 33\delta \lt 1[/math] (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 [math]H(k)[/math] for [math]k[/math] up to 5000). 
Jun 22  

Slight improvement in the [math]\tilde \theta[/math] parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of [math]k_0[/math]  
Jun 23  1,466 (Paldi/Harcos)  12,006 (Engelsma)  An improved monotonicity formula for [math]G_{k_01,\tilde \theta}[/math] reduces [math]\kappa_3[/math] somewhat  
Jun 24  [math](134 + \tfrac{2}{3}) \varpi + 28\delta \le 1[/math]? (v08ltu)
[math]140\varpi + 32 \delta \lt 1[/math]? (Tao)

1,268? (v08ltu)  10,206? (Engelsma)  A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates 
Jun 25  [math]116\varpi+30\delta\lt1[/math]? (FouvryKowalskiMichelNelson/Tao)  1,346? (Hannes)
1,007? (Hannes) 
10,876? (Engelsma)  Optimistic projections arise from combining the GrahamRingrose numerology with the announced FouvryKowalskiMichelNelson results on d_3 distribution 
Jun 26  [math]116\varpi + 25.5 \delta \lt 1[/math]? (Nielsen)
[math](112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta \lt 1[/math]? (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  [math]108\varpi + 30 \delta \lt 1[/math]? (Tao)  902? (Hannes)  6,966? (Engelsma)  Improved the Type III estimates by averaging in [math]\alpha[/math]; also some slight improvements to the Type II sums. Tuples page is now accepting submissions. 
Jul 1  [math](93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta \lt 1[/math]? (Tao) 
873? (Hannes)

Refactored the final CauchySchwarz in the Type I sums to rebalance the offdiagonal and diagonal contributions  
Jul 5  [math] (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta \lt 1[/math] (Tao) 
Weakened the assumption of [math]x^\delta[/math]smoothness of the original moduli to that of double [math]x^\delta[/math]dense divisibility  
Jul 10  7/600? (Tao)  An in principle refinement of the van der Corput estimate based on exploiting additional averaging  
Jul 19  [math](85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta \lt 1[/math]? (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  [math]168\varpi + 48\delta \lt 1[/math]# (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  [math]83\frac{1}{13}\varpi + 25\frac{5}{13} \delta \lt 1[/math]? (Tao)  603? (xfxie)  4,422?(Engelsma)
12 [EH] (Maynard) 
Worked out the dependence on [math]\delta[/math] 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#? (ClarkJarvis)  With this value of [math]k_0[/math], the value of [math]H[/math] given is best possible (and similarly for smaller values of [math]k_0[/math])  
Nov 19  105# (Maynard)
5 [EH] (Maynard) 
600# (Maynard/ClarkJarvis)  One also gets three primes in intervals of length 600 if one assumes ElliottHalberstam  
Nov 20 


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]) 42,392 [m=2]? (Nielsen) 
356?? (ClarkJarvis)  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 484,260 [m=2]? (Sutherland) 493,458 [m=2]#? Sutherland 

Nov 24  484,234 [m=2]? (Sutherland)
493,442 [m=2]#? (Sutherland) 484,192 [m=2]? (Sutherland) 

Nov 25  385#*? (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)
576# (ClarkJarvis) 
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#? (ClarkJarvis)  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#? (ClarkJarvis)
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 [math]M_k \geq \frac{k}{k1} \log k  O(1)[/math] lower bound.  
Dec 20  55#? (Nielsen) 36,000? [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)
272#? (ClarkJarvis) 

Dec 21  1,640,042? [m=3] (Sutherland)
1,631,027? [m=3] (Sutherland)

429,798? [m=2] (Sutherland)
25,602,438? [m=3] (Sutherland) 405,528? [m=2] (Sutherland)
25,533,684? [m=3] (Sutherland) 395,264? [m=2] (Sutherland) 395,178? [m=2] (Sutherland) 25,527,718? [m=3] (Sutherland)
24,490,758? [m=3] (Sutherland) 
Optimising the explicit lower bound [math]M_k \geq \log kO(1)[/math]  
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) 
395,154? [m=2] (Sutherland)
24,490,410? [m=3] (Sutherland)
395,122? [m=2] (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 
Legend:
 ?  unconfirmed or conditional
 ??  theoretical limit of an analysis, rather than a claimed record
 *  is majorized by an earlier but independent result
 #  bound does not rely on Deligne's theorems
 [EH]  bound is conditional the ElliottHalberstam conjecture
 [m=N]  bound on intervals containing N+1 consecutive primes, rather than two
 strikethrough  values relied on a computation that has now been retracted
See also the article on Finding narrow admissible tuples for benchmark values of [math]H[/math] for various key values of [math]k_0[/math].