If this hypothesis is correct, that would mean that when you’re 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 — beyond that point, you can’t calculate the terms in your sum. In the mid-1900s, number theorists proved many sieve theorems of the form, “If the generalized Riemann hypothesis is correct, then … ”
But a lot of these results didn’t actually need the full strength of the generalized Riemann hypothesis — 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 and Askold Vinogradov separately managed to prove just that: The primes have a level of distribution of at least 1/2, if we’re content with knowing that the buckets even out for almost every divisor.
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. “It’s kind of the gold standard of distribution theorems,” Tao said.
But mathematicians have long suspected — and numerical evidence has suggested — that the true level of distribution of the primes is much higher. In the late 1960s, Peter Elliott and Heini Halberstam conjectured that the level of distribution of the primes is just a shade below 1 — in other words, if you’re 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’re doing inclusion/exclusion, since they come up when you’re 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 “the dream.”
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 “square-root barrier” for prime numbers. This barrier, Lichtman said, is “a fundamental kind of waypoint in our understanding of the primes.”
New World Records
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 — in other words, whether the prime falls into the “2” bucket for any of these divisors. So you don’t need to know whether primes are evenly distributed across all the buckets for these divisors — you just need to know whether each “2” bucket holds the number of primes we expect.
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 by Bombieri, Friedlander and Henryk Iwaniec 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.
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 leap in mathematicians’ understanding of Fermat’s Last Theorem, which says that the equation an + bn = cn has no natural-number solutions for any exponent n higher than 2. (This was later proved in 1994 using techniques that didn’t 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.
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’s 1/2 level of distribution in a context where you’re sieving only with “smooth” numbers — ones that have no large prime factors. This tiny improvement enabled Zhang to prove the long-standing conjecture that as you go out along the number line, you’ll keep encountering pairs of primes that are closer together than some fixed bound. (Subsequently, Maynard and Tao each separately came up with another proof of this theorem, by using an improved sieve rather than an improved level of distribution.)
Zhang’s 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 “somewhat magical connection” 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 “the high-powered end of number theory.”
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 “tour de force,” 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.
Now, Maynard’s students are pushing these techniques further. Lichtman recently figured out how to extend Maynard’s 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’s the first time anyone has been able to use a level of distribution beyond the 1/2 from the classic Bombieri-Vinogradov theorem.
Another of Maynard’s students, Alexandru Pascadi, has matched the 0.617 figure 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.
Meanwhile, a third student, Julia Stadlmann, has boosted the level of distribution 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 in this context, reaching a 0.5017 level of distribution, and then an online collaboration called a Polymath project raised that number to 0.5233; Stadlmann has now raised it to 0.525.
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. “It’s like the 100-meter dash or something, [where] you shave 3.96 seconds to 3.95 seconds,” he said. Each new world record is “a benchmark for how much your methods have progressed.”
Overall, “the techniques are getting more clear and more unified,” he said. “It’s becoming clear, once you have an advance on one problem, how to adapt it to another problem.”
There’s no bombshell application for these new developments yet, but the new work “definitely changes the way we think,” Granville said. “This isn’t just banging a nail in harder — this is actually getting a more upgraded hammer.”
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