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(This message was added in version 6.7.0.) in /home4/scienrds/scienceandnerds/wp-includes/functions.php on line 6114Source:https:\/\/www.quantamagazine.org\/a-new-generation-of-mathematicians-pushes-prime-number-barriers-20231026\/#comments<\/a><\/br> If this hypothesis is correct, that would mean that when you\u2019re sieving up to 1 trillion, you can cross off multiples of 2, then 3, then 5, and keep going until the inclusion\/exclusion sum starts to involve divisors over about 1 million \u2014 beyond that point, you can\u2019t calculate the terms in your sum. In the mid-1900s, number theorists proved many sieve theorems of the form, \u201cIf the generalized Riemann hypothesis is correct, then \u2026 \u201d<\/p>\n But a lot of these results didn\u2019t actually need the full strength of the generalized Riemann hypothesis \u2014 it would be enough to know that primes were well distributed into buckets for almost every divisor, instead of every single divisor. In the mid-1960s, Enrico Bombieri<\/a> and Askold Vinogradov separately<\/a> managed<\/a> to prove just that: The primes have a level of distribution of at least 1\/2, if we\u2019re content with knowing that the buckets even out for almost every divisor.<\/p>\n The Bombieri-Vinogradov theorem, which is still widely used, instantly proved many of the results that had previously relied on the unproved generalized Riemann hypothesis. \u201cIt\u2019s kind of the gold standard of distribution theorems,\u201d Tao said.<\/p>\n But mathematicians have long suspected \u2014 and numerical evidence has suggested \u2014 that the true level of distribution of the primes is much higher. In the late 1960s, Peter Elliott<\/a> and Heini Halberstam conjectured<\/a> that the level of distribution of the primes is just a shade below 1 \u2014 in other words, if you\u2019re looking at primes up to some huge number, they should be evenly distributed into buckets even for divisors very close in size to the huge number. And these large divisors matter when you\u2019re doing inclusion\/exclusion, since they come up when you\u2019re correcting for overcounts. So the closer mathematicians can get to the level of distribution Elliott and Halberstam predicted, the more terms they can calculate in the inclusion\/exclusion sum. Proving the Elliott-Halberstam conjecture, Tao said, is \u201cthe dream.\u201d<\/p>\n To this day, however, no one has been able to beat the 1\/2 level of distribution in the full degree of generality that the Bombieri-Vinogradov theorem achieves. Mathematicians have taken to calling this stumbling block the \u201csquare-root barrier\u201d for prime numbers. This barrier, Lichtman said, is \u201ca fundamental kind of waypoint in our understanding of the primes.\u201d<\/p>\n For many sieve problems, though, you can make progress even with incomplete information about how the primes divide into buckets. Take the twin primes problem: Sieving out a prime if the number two spots to its left is divisible by 3 or 5 or 7 is the same as asking whether the prime itself has a remainder of 2 when divided by 3 or 5 or 7 \u2014 in other words, whether the prime falls into the \u201c2\u201d bucket for any of these divisors. So you don\u2019t need to know whether primes are evenly distributed across all the buckets for these divisors \u2014 you just need to know whether each \u201c2\u201d bucket holds the number of primes we expect.<\/p>\n In the 1980s, mathematicians started figuring out how to prove distribution theorems that focus on one particular bucket. This work culminated in a 1986 paper<\/a> by Bombieri, Friedlander and Henryk Iwaniec<\/a> that pushed the level of distribution up to 4\/7 (about 0.57) for single buckets, not for all sieves but for a wide class of them.<\/p>\n As with the Bombieri-Vinogradov theorem, the body of ideas developed in the 1980s found a host of applications. Most notably, it enabled a huge<\/a> leap<\/a> in mathematicians\u2019 understanding of Fermat\u2019s Last Theorem, which says that the equation an<\/sup><\/em> + bn<\/sup><\/em> = cn<\/sup><\/em> has no natural-number solutions for any exponent n <\/em>higher than 2. (This was later proved in 1994 using techniques that didn\u2019t rely on distribution theorems.) After the excitement of the 1980s, however, there was little progress on the level of distribution of the primes for several decades.<\/p>\n Then in 2013, Zhang figured out how to get over the square-root barrier in a different direction from that of Bombieri, Friedlander and Iwaniec. He dug into old, unfashionable methods from the early 1980s to eke out the tiniest of improvements on Bombieri and Vinogradov\u2019s 1\/2 level of distribution in a context where you\u2019re sieving only with \u201csmooth\u201d numbers \u2014 ones that have no large prime factors. This tiny improvement enabled Zhang to prove the long-standing conjecture<\/a> that as you go out along the number line, you\u2019ll keep encountering pairs of primes that are closer together than some fixed bound. (Subsequently, Maynard and Tao each separately came up with<\/a> another proof of this theorem, by using an improved sieve rather than an improved level of distribution.)<\/p>\n Zhang\u2019s result drew on a version of the Riemann hypothesis that lives in the world of algebraic geometry. The work of Bombieri, Friedlander and Iwaniec, meanwhile, relied on what Maynard calls a \u201csomewhat magical connection\u201d to objects called automorphic forms, which have their own version of the Riemann hypothesis. Automorphic forms are highly symmetric objects that, Tao says, belong to \u201cthe high-powered end of number theory.\u201d<\/p>\n A few years ago, Maynard became convinced that it should be possible to squeeze more juice out of these two methods by combining their insights. In his series of three papers in 2020, which Granville labeled a \u201ctour de force,\u201d Maynard managed to push the level of distribution up to 3\/5, or 0.6, in a slightly narrower context than the one Bombieri, Friedlander and Iwaniec studied.<\/p>\n Now, Maynard\u2019s students are pushing these techniques further. Lichtman recently figured out<\/a> how to extend Maynard\u2019s level of distribution to about 0.617. He then parlayed this increase into new upper bounds on the counts of both twin primes and Goldbach representations of even numbers as the sum of two primes. For the latter, it\u2019s the first time anyone has been able to use a level of distribution beyond the 1\/2 from the classic Bombieri-Vinogradov theorem.<\/p>\n Another of Maynard\u2019s students, Alexandru Pascadi<\/a>, has matched the 0.617 figure<\/a> for the level of distribution not of primes but of smooth numbers. Like primes, smooth numbers come up all over number theory, and results about their level of distribution and that of the primes often go hand in hand.<\/p>\n Meanwhile, a third student, Julia Stadlmann<\/a>, has boosted the level of distribution<\/a> of primes in the setting that Zhang studied, in which the divisors (instead of the numbers being divided) are smooth numbers. Zhang narrowly beat the square-root barrier<\/a> in this context, reaching a 0.5017 level of distribution, and then an online collaboration called a Polymath project raised that number<\/a> to 0.5233; Stadlmann has now raised it to 0.525.<\/p>\n Other mathematicians tease analytic number theorists, Tao said, for their obsession with small numerical advances. But these tiny improvements have a significance beyond the numbers in question. \u201cIt\u2019s like the 100-meter dash or something, [where] you shave 3.96 seconds to 3.95 seconds,\u201d he said. Each new world record is \u201ca benchmark for how much your methods have progressed.\u201d<\/p>\n Overall, \u201cthe techniques are getting more clear and more unified,\u201d he said. \u201cIt\u2019s becoming clear, once you have an advance on one problem, how to adapt it to another problem.\u201d<\/p>\n There\u2019s no bombshell application for these new developments yet, but the new work \u201cdefinitely changes the way we think,\u201d Granville said. \u201cThis isn\u2019t just banging a nail in harder \u2014 this is actually getting a more upgraded hammer.\u201d<\/p>\n Quanta\u00a0is conducting a series of surveys to better serve our audience. Take our\u00a0<\/i>mathematics reader survey<\/i><\/a>\u00a0and you will be entered to win free\u00a0<\/i>Quanta\u00a0merch.<\/i><\/p>\n<\/div>\n <\/br><\/br><\/br><\/p>\n
\nA New Generation of Mathematicians Pushes Prime Number Barriers<\/br>
\n2023-10-30 21:58:25<\/br><\/p>\nNew World Records<\/strong><\/h2>\n