I’ll be very annoyed if this result doesn’t get published before the end of the year.

@GHfromMO: Had he worked harder, he would have been able to publish it last year, with the better exponent of $-2021$.

@AlexM. Breaking news: Polymath project invents time machine to improve another one of Zhang’s bounds.

This new range is sufficient for analytic number theorists to apply it to number theory problems and obtain a large number of meaningful results. In many previous papers, it is assumed that Siegel zero does not exist, but it does not need to be so strong. Zhang's paper did not rule out the possibility of the existence of Siegel zeros, but the scope of this exclusion is enough to cover the scope required by many previous papers. This makes many previous results from hypothetical results to deterministic results. From this perspective, the significance of the paper is extremely significant

Assuming the paper is correct, an immediate consequence is that every ineffective constant stemming from Siegel's theorem is now effective. Of course, the result implies more than this, but that is already enough for the result to be monumental if correct.

I am very amazed by how elementary this paper is...

@NoOne amazed in a good way?

@DavidRoberts well I haven't read it in details but my first impression is just that it is a very high level Kong-Fu of hard analysis rather than Zen of modern number theory...

@DavidRoberts and I should add that in mathematics I appreciate both "Kong-Fu" and "Zen" although oftentimes "Kong-Fu" looks cooler :p

@TanmayKhale Only if Zhang's constants are effective.

@GHfromMO Doesn't the abstract explicitly say Zhang's constants are effective?

@Carl-FredrikNybergBrodda Surely it does. I am away from the paper now, so I cannot check.

@GHfromMO Indeed, the abstract states "It is proved that L(1,chi) >> (log D)^(-2022), where the constant is absolute and effectively computable." This was the first thing I checked! =)

@GHfromMO The two main theorems (Theorems 1 and 2, lower bound for L(1,chi), zero-free region for L(s,chi), respectively) also say "absolute and effectively computable." Of course the deduction of the latter from the former is a standard + effective MVT argument.

Thé question i would Ask what thèse important results add to Riemann hypothesis ? Does thèse results cany Say anythings about validity of RH ?

@zeraouliarafik no.

@Blanco Are you really sure that this is the version that's going to be on arxiv? I could believe that Zhang wrote this though, but I could also believe that this is an earlier version with some substantial errors that he had fixed by now.

@zeraouliarafik This doesn't directly say anything about RH, except in so far as if you care about two conjectures A and B, and you know that A implies B, and then you prove B, that should make you assign a higher credence to A than you did previously.

A version has now appeared on the arxiv arxiv.org/abs/2211.02515 (it was posted on friday 4th)

@JamesLeng I was not sure. I believe it is not the final submitted version, but as far as I know, it is the latest version that can be found on the internet.

someone here can know how likely he could prove it right?

@HollisWilliams, that's overly sensitive I think. And it was "Kong Fu"!

@kodlu It's the same thing: gōng fu or kōng fu.

We should wait what Terry Tao say about these important results because Tao works on the same domain and his opinion is very important than these obtained results.Just guess