Will Sawin on Peter Woit’s blog points out that Zhang is making a weaker claim than non-existence of Siegel zeros:
> The “no Siegel zeros” conjecture is that the distance of any real zero of L(s,chi_D) from 1 is bounded below by a constant times 1/log D. The result Yitang Zhang claims is that this distance is bounded below by a constant times 1/(log D)^2024. So both involve the claim that there are no zeroes too close to the point 1, but one is stronger than the other by a large power.
> However, Yitang Zhang’s stated result is a big improvement on what was known before – the only effective estimates had a lower bound for the distance of something close to 1/sqrt(D).
> The situation is very analogous to the twin primes paper where he improved the bound from the previously known infinity to 70,000,000, and it was expected that optimization of the method would rapidly get it down further, though falling short of the conjectural value of 2.
Out of curiosity, and asking as someone who didn't do much math beyond university algebra and calculus... how do seemingly arbitrary constants like "70 million" show up in what I imagine is a theory-dense proof?
Let’s say you have a complex expression, say with a few gnarly infinite sums and want to know its value.
Typically, your first step typically is to show that it is finite because that often is a lot easier, and if it turns out not being finite, you’ve spared yourself from doing some complex calculations.
The typical approach to show that some expression is finite is to replace difficult parts of the expression by simpler ones that certainly aren’t smaller. If the changed, definitely not smaller, expression is finite, so must be the original one.
As a simple example, a rouge estimate of sin(x) + cos(x) says both sin(x) and *cos(x) are in [-1,1], so their sum is in [-2,2]. That shows the expression is finite for all x. (Actually, the sum is in [-√2, √2], but if you don’t need to be that precise, why bother?)
Similarly, I wouldn’t know what the sum of i=1 to infinity of |sin(i)|/2^i is, but each term is positive and not greater than 1/2^i, so the sum isn’t greater than 1, and that series must converge.
In addition to what the sibling commenters wrote --
The basic strategy of Zhang, as well as Goldston-Pintz-Yildirim, Maynard, and others is as follows: consider the numbers
n, n+2, n+4, n+6, n+8, ... n+70,000,000
and prove that "on average" at least two of them are prime. Prove this, and the result follows.
Now this is not literally true, the average spacing between primes is unbounded. The main technical work is in coming up with a weighting function, which is biased towards those n for which the "average" claim is likely to be true. Prove this, and you're done.
One then has to compute "the weighted average number of primes in n, n+2, ..., n+70,000,000", subject to the weighting function you chose. It's relatively easy to do in principle, but it requires various estimates for counting prime numbers, all of which come with error terms, and the hard part is to prove that the error terms don't accumulate enough to completely drown out the main term.
Work on this subject has proceeded along two lines: (1) come up with a more efficient weighting function, that does a better job of picking out the n we really want to count; (2) improving the error estimates in the various counts that come up. Both of these are intrinsically quantitative, and inevitably less precise than one expects to be true -- and the upshot is that if you aim for 2, you might prove 70,000,000 instead, which is still an amazing achievement.
My understanding is that 70 million was indeed chosen arbitrarily, but it is large enough to make the proof work without too much extra work. The "2022" number in the recent claimed proof might also be in this vein, i.e., there is nothing special about it, it's just a number large enough to make the proof succeed relatively comfortably, and probably also a play on the current year.
It depends on the length of the paper, the quality of the writing, and if there is an error near the beginning. Mochizuki's ABC papers had problems near the beginning so that's why at least some problems were found rather quickly, and it also went "viral" in the community so more people looked at it.
However, if this proof is sound, well-written, and is being looked at publicly by the top number theorists then it could be validated in under a year. Actually that's pretty much the same time it takes to peer-review a lot of ten page papers too that are not in the news...
I personally think media reports of most academic results is a distraction. Experts would surely start to weigh in very soon if this result is as significant as claimed. It doesn't hurt to wait a bit before discussing it. As the story of Zhang is an inspiring one, I don't fault mass media from jumping on it. However, I don't think a serious scientific journal like Nature should be doing this.
Nature has news department and this is news, not article. Nature News is in fact very good if you evaluate it as a science news, not a science journal (which it isn't).
> The “no Siegel zeros” conjecture is that the distance of any real zero of L(s,chi_D) from 1 is bounded below by a constant times 1/log D. The result Yitang Zhang claims is that this distance is bounded below by a constant times 1/(log D)^2024. So both involve the claim that there are no zeroes too close to the point 1, but one is stronger than the other by a large power.
> However, Yitang Zhang’s stated result is a big improvement on what was known before – the only effective estimates had a lower bound for the distance of something close to 1/sqrt(D).
> The situation is very analogous to the twin primes paper where he improved the bound from the previously known infinity to 70,000,000, and it was expected that optimization of the method would rapidly get it down further, though falling short of the conjectural value of 2.
https://www.math.columbia.edu/~woit/wordpress/?p=13137#comme...