Monumental (if correct) advance in number theory posted to ArXiv by Yitang Zhang
I can&#x27;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 &#x2F; 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 &quot;unless a Siegel zero exists,&quot; or split into two cases based on if Siegel zeros exist or don&#x27;t exist, etc.<p>But here is the worst (or &quot;most mysterious,&quot; depending on your mood..) thing about Siegel zeros. Our best result about Siegel zeros (excluding for present discussion Zhang&#x27;s work), namely Siegel&#x27;s theorem, is ineffective. That is, it says &quot;there exists some constant C &gt; 0 such that...&quot; but it can tell you nothing about that constant beyond that it is positive and finite (we say that the constant is &quot;not effectively computable from the proof&quot;).*<p>So then, if you try to use Siegel&#x27;s theorem to prove things about primes, this ineffectivity trickles down (think &quot;fruit of the poisoned tree&quot;). 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. <i>However</i>, the proof in most standard texts fundamentally <i>cannot</i> tell you what the threshold for &quot;sufficiently large&quot; is, because the proof uses Siegel&#x27;s theorem! In this particular case, it turns out that one can avoid Siegel&#x27;s theorem, and in fact the statement &quot;Any odd integer larger than <i>five</i> is the sum of three primes&quot; is now known <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Goldbach's_weak_conjecture" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Goldbach's_weak_conjecture</a>. But it is certainly not always possible to avoid Siegel&#x27;s theorem, and Zhang&#x27;s result would make so many theorems which right now involve ineffectively computable constants effective.<p>*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&#x27;t occur if GRH is true. Next, assume GRH is false. Take a &quot;minimal&quot; counterexample to GRH, and use it to &quot;repel&quot; or &quot;exclude&quot; other possible counterexamples.