Number theorist here. This is a massive big deal, and would likely be a Fields Medal for a human if a human had done it. But it is an exaggeration to say it is the biggest result in 200 years. At a minimum, it is hard to argue that it is a bigger result than the proof of the prime number theorem in 1896 (which this is a strengthening of), or Riemann's original 1859 paper where he laid out the zeta function and its analytic importance, or Dirichlet's proof of infinitely many primes in arithmetic progressions which is the late 1830s.
But yeah, this is still a very big deal. Among other things, it will drastically improve all sorts of Rosser-Schoenfeld type results for the PNT and that's just a start. For comparison, I have a paper form 2018 where this result would cut 3 pages out and make the full result cleaner and much tighter, and there are likely hundreds of papers like this.
I am also an analytic number theorist, and I disagree. Not only do I think Fields Medal is an understatement (Fields Medals have been awarded for far less than proving quasi-RH + no Siegel zeros), I don't think it is unfair to say that this is a bigger deal than the 1896 proof of the PNT.
As for Riemann's memoir, it's hard to compare. You could argue that was "just" noticing a connection (between number theory and Fourier analysis) that nobody had noticed before; in fact this is the kind of thing AI is extremely good at. I'm being a little cute here.
I think if a human had proven just these two results in the form of a uniform zero-free region for L(s,chi) from nothing as OpenAI did it would not be unfair to say that it would be the single greatest advance in math (easily dwarfing Wiles' FLT), and it would instantly put them in the ranks of greatest mathematicians of all time. Unlike something like Navier Stokes there wasn't a semblance of a research program, experts basically considered this hopeless and would have said the chance of seeing a proof in our lifetime was near zero.
For some comparison, Yitang Zhang's bounded gaps result might have gotten him a Fields Medal if he was not disqualified by age. When it was floated that he might have proven Siegel zeros don't exist, it was considered (by experts) clearly a much bigger deal. This result blows that out of the water (it's a way better version); at least analytic number theorists I talked to thought it was plausible but unlikely that Siegel zeros would be eliminated in our lifetime but thought RH was basically hopeless.
While some of my work is in analytic number theory, much is in other subareas, so it is possible I should defer to you on this.
It seems to me less than PNT in terms of what can we actually do with this. Many different areas of math use PNT, and from my standpoint, PNT is helpful not just for what it implies directly but because it lets us make really good heuristics about whether some sets are infinite or not, and what their rough size is. (Granted, one can do that also mostly via Chebyshev). For those purposes, this doesn't really enter in. Similarly, PNT feels like a statement at least I can say explain to my mother without any technical details. This isn't that. But that may also be my own biases of wanting things to cash out to very concrete statements about the integers.
I agree that one striking element is how no one saw this coming. This isn't building on an existing research program, which itself is remarkable. And last night, before I went to bed, I saw a conversation between a bunch of analytic number theorists who seemed to think there was potentially some slack in the quasi-RH argument, which if that's the case means this is going to go even further.
Thank you for the detailed explanation. From what I'm reading from a lot of mathematicians there's at least a dozen of results here that are field-definining and worthy at minimum of a Fields medal.
I guess the biggest news are not the discoveries themselves but how they were found and that math is going through the biggest revolution as a field since almost ever.
1896 PNT is basically 1859 Riemann + a trig inequality.
1830 Dirichlet's result is qualitative only, it shows infinitude but not the asymptote in terms of the zeros for it predates Riemann.
To me this is the first substantial step after the 1896 PNT, and we really do not see much progress in the whole 20th century.
Personally so far there are only two people worth mentioning,
- Euler, introduces the real zeta function and Euler product, establishes the functional equation at (half?) integers.
- Riemann, introduces complex analysis ideas to the zeta function.
And of course this result if it is true. This is first to penetrate the critical strip, which nobody had any idea how to approach for over a century and a half.
Number theorist here. This is a massive big deal, and would likely be a Fields Medal for a human if a human had done it. But it is an exaggeration to say it is the biggest result in 200 years. At a minimum, it is hard to argue that it is a bigger result than the proof of the prime number theorem in 1896 (which this is a strengthening of), or Riemann's original 1859 paper where he laid out the zeta function and its analytic importance, or Dirichlet's proof of infinitely many primes in arithmetic progressions which is the late 1830s.
But yeah, this is still a very big deal. Among other things, it will drastically improve all sorts of Rosser-Schoenfeld type results for the PNT and that's just a start. For comparison, I have a paper form 2018 where this result would cut 3 pages out and make the full result cleaner and much tighter, and there are likely hundreds of papers like this.
I am also an analytic number theorist, and I disagree. Not only do I think Fields Medal is an understatement (Fields Medals have been awarded for far less than proving quasi-RH + no Siegel zeros), I don't think it is unfair to say that this is a bigger deal than the 1896 proof of the PNT.
As for Riemann's memoir, it's hard to compare. You could argue that was "just" noticing a connection (between number theory and Fourier analysis) that nobody had noticed before; in fact this is the kind of thing AI is extremely good at. I'm being a little cute here.
I think if a human had proven just these two results in the form of a uniform zero-free region for L(s,chi) from nothing as OpenAI did it would not be unfair to say that it would be the single greatest advance in math (easily dwarfing Wiles' FLT), and it would instantly put them in the ranks of greatest mathematicians of all time. Unlike something like Navier Stokes there wasn't a semblance of a research program, experts basically considered this hopeless and would have said the chance of seeing a proof in our lifetime was near zero.
For some comparison, Yitang Zhang's bounded gaps result might have gotten him a Fields Medal if he was not disqualified by age. When it was floated that he might have proven Siegel zeros don't exist, it was considered (by experts) clearly a much bigger deal. This result blows that out of the water (it's a way better version); at least analytic number theorists I talked to thought it was plausible but unlikely that Siegel zeros would be eliminated in our lifetime but thought RH was basically hopeless.
While some of my work is in analytic number theory, much is in other subareas, so it is possible I should defer to you on this.
It seems to me less than PNT in terms of what can we actually do with this. Many different areas of math use PNT, and from my standpoint, PNT is helpful not just for what it implies directly but because it lets us make really good heuristics about whether some sets are infinite or not, and what their rough size is. (Granted, one can do that also mostly via Chebyshev). For those purposes, this doesn't really enter in. Similarly, PNT feels like a statement at least I can say explain to my mother without any technical details. This isn't that. But that may also be my own biases of wanting things to cash out to very concrete statements about the integers.
I agree that one striking element is how no one saw this coming. This isn't building on an existing research program, which itself is remarkable. And last night, before I went to bed, I saw a conversation between a bunch of analytic number theorists who seemed to think there was potentially some slack in the quasi-RH argument, which if that's the case means this is going to go even further.
Thank you for the detailed explanation. From what I'm reading from a lot of mathematicians there's at least a dozen of results here that are field-definining and worthy at minimum of a Fields medal.
I guess the biggest news are not the discoveries themselves but how they were found and that math is going through the biggest revolution as a field since almost ever.
> it would be the single greatest advance in math
Did you mean to not qualify that? That is a bold statement indeed.
1896 PNT is basically 1859 Riemann + a trig inequality.
1830 Dirichlet's result is qualitative only, it shows infinitude but not the asymptote in terms of the zeros for it predates Riemann.
To me this is the first substantial step after the 1896 PNT, and we really do not see much progress in the whole 20th century. Personally so far there are only two people worth mentioning,
- Euler, introduces the real zeta function and Euler product, establishes the functional equation at (half?) integers.
- Riemann, introduces complex analysis ideas to the zeta function.
And of course this result if it is true. This is first to penetrate the critical strip, which nobody had any idea how to approach for over a century and a half.
How about 100 years?
Yeah, completely reasonable to argue that.
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(unrelated: love your username)