[gavagai691]

I can't resist saying one last thing about Siegel zeros: number theorists REALLY would like for this result to be correct because the possibility of Siegel zeros is unbelievably annoying. I mean mathematicians are supposed to enjoy challenges / difficulties, but Siegel zeros are just so recurrently irritating. The possibility of Siegel zeros means that in so many theorems you want to write down, you have to write caveats like "unless a Siegel zero exists," or split into two cases based on if Siegel zeros exist or don't exist, etc.

But here is the worst (or "most mysterious," depending on your mood..) thing about Siegel zeros. Our best result about Siegel zeros (excluding for present discussion Zhang's work), namely Siegel's theorem, is ineffective. That is, it says "there exists some constant C > 0 such that..." but it can tell you nothing about that constant beyond that it is positive and finite (we say that the constant is "not effectively computable from the proof").*

So then, if you try to use Siegel's theorem to prove things about primes, this ineffectivity trickles down (think "fruit of the poisoned tree"). For example, standard texts on analytic number theory include a proof of the following theorem: any sufficiently large odd integer is the sum of three primes. However, the proof in most standard texts fundamentally cannot tell you what the threshold for "sufficiently large" is, because the proof uses Siegel's theorem! In this particular case, it turns out that one can avoid Siegel's theorem, and in fact the statement "Any odd integer larger than five is the sum of three primes" is now known https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture. But it is certainly not always possible to avoid Siegel's theorem, and Zhang's result would make so many theorems which right now involve ineffectively computable constants effective.

*Why is the constant not effectively computable? Because the proof proceeds basically like this. First: assume the Generalized Riemann Hypothesis. Then the result is trivial, Siegel zeros are exceptions to GRH and don't occur if GRH is true. Next, assume GRH is false. Take a "minimal" counterexample to GRH, and use it to "repel" or "exclude" other possible counterexamples.

[SantalBlush]

>I can't resist saying one last thing

Please, keep going. This is good reading.

[gavagai691]

In that case, you might find interesting these two short explanations I posted to Reddit about Siegel zeros (the second is a continuation of the first) :)

https://old.reddit.com/r/math/comments/y93a86/eliundergradua...

https://old.reddit.com/r/math/comments/ymlacu/professor_yita...

The class number formula, mentioned in the second comment, is one of the craziest "bridge results" in all of math (meaning a result that connects two seemingly disparate areas). The class number formula connects the values of Dirichlet L-functions at s = 1 (Dirichlet L-functions are complex functions related to the distribution of primes in arithmetic progressions), to class numbers of number fields. (Remember that the value of Dirichlet L-functions at 1 is exactly what the question of Siegel zeros concerns.)

To give a crash course on what some of those words mean:

1. A number field is what you get when you take the rational numbers, and you throw in the roots of some polynomials to get a bigger object where you can still do all of the usual arithmetic operations, in the same way that we throw in the roots of x^2 + 1 (namely, i, -i) into the real numbers to get the complex numbers.

2. The ring of integers is the right notion of the "integers" in that number field. (That is, rational numbers : integers = number field : ring of integers in that number field.)

3. The class number of a number field tells you how close you are to having unique factorization into primes holding in the ring of integers of that number field*. If the class number is 1, then you have unique factorization; if the class number is 1000, then you are very far from it.

What this connections means is that you can prove things about regular old primes in arithmetic progressions (in the integers) by proving things about these exotic / abstract primes (in rings of integers of number fields), and vice-versa.

Anyway, as a result of the class number formula, there are a lot of results about class numbers that are ineffective because of Siegel's theorem too, e.g., https://en.wikipedia.org/wiki/Brauer%E2%80%93Siegel_theorem. Zhang's result (if correct) would make all of those effective, too.

*While in the integers, it is true that every number factors uniquely into a product of primes, this is unfortunately not true in more general contexts. In fact, algebraic number theory basically began with a mistaken proof of Fermat's Last Theorem, which was mistaken precisely because it assumed that unique factorization always holds in this more general context, which is not true. (If unique factorization did always hold, then that proof of FLT would have been correct.)