Timeline for Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
Current License: CC BY-SA 4.0
23 events
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| Mar 12, 2019 at 20:09 | comment | added | Zach Teitler | This answer has been edited so that equation $(*)$ is quantified over a variable $C$, but there is no $C$ in $(*)$. Is it supposed to be for all $x > C$? | |
| Mar 12, 2019 at 12:00 | history | edited | Solomon Ucko | CC BY-SA 4.0 |
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| Jan 30, 2015 at 5:06 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| Jun 2, 2013 at 0:36 | comment | added | Kim Morrison | Over at sbseminar.wordpress.com/2013/05/30/… we've been slowly scraping away the slack. We got the bound down to 42,342,946, but there's very little in the way of cheap improvements left now. | |
| May 24, 2013 at 5:53 | comment | added | pageman | @Mark Zhang started with 70,000,00 & via your comment it's now 63,374,611 - could Zhang have started with a lower number? | |
| May 21, 2013 at 22:49 | history | edited | Alon Amit | CC BY-SA 3.0 |
Trying to fix the first two links to Wikipedia
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| May 21, 2013 at 21:05 | comment | added | Denis Chaperon de Lauzières | After a first look at the paper, I actually have the impression that Zhang does not use automorphic results or techniques, but only some of the dispersion method / combinatorial ideas which occur as ingredients (together with automorphic results) in the works of Bombieri-Fouvry-Friedlander-Iwaniec... | |
| May 20, 2013 at 11:09 | vote | accept | pageman | ||
| May 20, 2013 at 10:24 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 9:55 | comment | added | Mark Lewko | @Ben, I have no idea which approach Zhang follows, I haven't seen a copy of the paper. I see your right that I wasn't looking at the table associated to the more refined argument. I'm curious to what extent the length of admissible tuples of a given size have been nailed down (note that the table on page 9 of the GPY paper seems to suggest the smallest tuple of size 421 isn't known). It might be possible to reduce the bound by a non-negligible factor just by computing a smaller admissible tuple of whatever size Zhang needs. | |
| May 20, 2013 at 9:45 | comment | added | Ben Green | Mark - we added our comments at the same time. I was referring to the table on page 12 of GPY, whereas you were referring to the one on p9. I guess the one on page 9 tells you what you can get using just the basic GPY method, and the one on page 12 uses more complicated weights. I guess Zhang elaborates on the basic GPY method. Maybe his method can be combined with the more complicated GPY method which leads to the numerics on page 12; it seems likely that this will be one place any would-be 70000000-reducers will look first. | |
| May 20, 2013 at 9:39 | comment | added | Ben Green | It's interesting to speculate on how much the 70000000 will be reduced. In this regard, the table on page 12 of Goldston-Pintz-Yildirim is relevant. If one had level of distribution 4/7 with no strings attached, it seems one might get gaps of size 500 or so. To get gaps under 100 without some completely new idea one would have to go out to level of distribution nearly 2/3, i.e. double the improvement of BFI. I'd wager that getting down to 10000 or so is going to prove pretty difficult. | |
| May 20, 2013 at 9:36 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 8:54 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 8:48 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 8:13 | comment | added | Mark Lewko | @Denis, thanks for the correction. I've update the text to include a reference to the Fouvry-Iwaniec result. Certainly one of the most fascinating aspects of Zhang's work is the (likely) application of deep estimates from the theory of Kloosterman sums and automorphic forms via the Bombieri-Friedlander-Fouvry-Iwaniec-type arguments. This, of course, should be compared to the Goldston-Pintz-Yildirim estimates that rely on Bombieri-Vinogradov, which in turn is derived from more classical estimates (such as Siegel-Walfisz, Vaughn's identity and the large sieve). | |
| May 20, 2013 at 7:22 | history | edited | Mark Lewko | CC BY-SA 3.0 |
Included Fouvry-Iwaniec reference
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| May 20, 2013 at 7:11 | comment | added | Denis Chaperon de Lauzières | Note: the first result of "Bombieri-Friedlander-Iwaniec style", going beyond the Riemann Hypothesis but not as far as the exponent $4/7$, is due to Fouvry-Iwaniec ("Primes in arithmetic progressions", Acta Arith. 42 (1983), 197-218; the exponent there is $9/17$). All these results depend crucially on the spectral theory of automorphic forms to estimates sum of Kloosterman sums, and especially on the results of Deshouillers and Iwaniec in this direction. | |
| May 20, 2013 at 6:56 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 5:28 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 5:22 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 5:10 | history | edited | Mark Lewko | CC BY-SA 3.0 |
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| May 20, 2013 at 5:05 | history | answered | Mark Lewko | CC BY-SA 3.0 |