An experimental AI-math challenge: using tenth discriminant bounds to explore arguments ruling out Siegel zeroes

In the spirit of my recent “Math 2.0” talk on new ways to use AI assistance to enhance the human understanding of mathematics, rather than substitute for it, I am using this post to propose an experimental challenge, inspired by the recent activity around excluding Landau–Siegel zeroes, as discussed in this previous post.

There is a well-known connection between Landau–Siegel zeroes and the tenth discriminant problem, which I will briefly recall here; an in-depth discussion on this problem can be found in this survey of Shafarevich. As far back as the work of Gauss, it was observed that there were nine discriminants whose rings of integers for the associated quadratic field enjoyed unique factorization, namely

Gauss believed this list of nine to be the complete list of such discriminants (the “class number one problem”), but this was only proven by Heegner in 1952 by introducing the concept of a Heegner point on the modular curve; and the above numbers are now known as Heegner numbers.

Prior to Heegner’s solution was the work of Heilbronn and Linfoot in 1934, who used what we now call the Deuring–Heilbronn repulsion phenomenon to show that the existence of a tenth discriminant in the above list can be used to “repel” any further such discriminants, so that there was at most one further Heegner number beyond the nine already known. However, this argument was ineffective: no upper bound on this putative tenth discriminant was given, and so there was no obvious way to solve the class number one problem purely from this result.

There is a close connection to Landau–Siegel zeroes: if is a Heegner number, and is the quadratic character associated to this quadratic field, then the class number formula gives , and with a bit more work one can produce a Landau–Siegel zero with . For sufficiently large , this is inconsistent with Siegel’s theorem; but the standard proofs of this theorem also relied on the repulsion phenomenon and were also ineffective. It was only in 1976 that Goldfeld gave an effective version of Siegel’s theorem that was enough to at least bound the tenth discriminant; his original argument gave an upper bound of the form , but a subsequent refinement by Oesterlé improved this to .

But now we have a fair number of new (and effective) ways to rule out Siegel zeroes, either by establishing a quasi-Riemann hypothesis, or by some version of the argument in this previous post. In principle, each version of these arguments should also give a bound on the tenth discriminant (without using Heegner’s solution to the class number one problem that rules out this discriminant). The task of making each of these estimates explicit enough to create such a bound is rather tedious for humans. However, it seems eminently suitable for AI.

Thus, the challenge I pose is, for each of the new manuscripts or digestions relating to either the quasi-Riemann hypothesis or the Landau–Siegel zero problem (or by some novel argument), to use an AI tool to work out an effective upper bound for a hypothetical tenth discriminant , without using Heegner’s methods, and to make best efforts to optimize this constant. This serves several mathematical purposes:

• If the threshold is low enough to resolve the remaining discriminants by direct computation, this would yield a new solution of the class number one problem — possibly one that is formalizable in Lean in a high quality fashion (suitable for instance for upstreaming to Mathlib).
• By performing this exercise on each of the different writeups, we obtain a way to compare the “strength” of each such writeup (after performing whatever optimizations are available for each writeup), which could give insight as to the relative advantages or disadvantages of different arrangements of the methods.
• The exercise of minimizing the upper bound may spur the creation of entirely new arguments to solve either the class number one problem, or the elimination of Landau–Siegel zeroes.
• There are many previous “failed” attempts at excluding the Landau–Siegel zero. It is possible that some of them contain ideas that can be combined with the newer solutions to this problem to “repair” the proofs, or at least improve the bound on the tenth discriminant.
• Efforts here may also help improve the state of knowledge on higher class number problems, which by the work of Watkins in 2004 is currently solved up to class number 100.

Furthermore, as both of these problems are already solved (for over seven decades, in the case of the class number one problem), there is little risk of such a competition inadvertently disrupting an ongoing research project.

I am therefore inviting interested readers to contribute in the comments any effective bounds on the tenth discriminant they can extract from either an existing argument, or a novel one, with any amount of AI use permitted for this challenge so long as it is disclosed. Note that the challenge is broader than simply getting the “best” bound overall amongst all possible approaches; I am also interested in what bounds one can extract from any specific approach, even if they are worse than, say, the bounds of Goldfeld or Oesterlé. One could also explore other directions, such as bounds that avoid using extensive compute to squeeze out the maximal amount of optimization, but are instead clean enough that their arguments are human-verifiable; or to find arguments that are “elementary” in various senses, for instance using only tools available to an undergraduate mathematics student.

Any large outputs should be posted as links rather than as plain text; for instance, pointing to a Github repository containing one’s work would be a reasonable way to submit an entry.

I would ask contributors to not prioritize speed above quality; this is not intended to be a race regarding who is “first” to claim any given bound. The spirit is intended to be collaborative rather than competitive: think of a Polymath style project, updated for the AI era. In particular, as with traditional mathematical research, it would be appropriate to build upon previous contributions to obtain an improved bound for a given proof technique. (But in such cases, one should of course give appropriate attribution to the prior source.)

I will not insist on formal verification (e.g., in Lean) of the results, though I would ask anyone submitting a purely AI-generated solution to do enough due diligence to check the output (e.g., with an independent AI tool) before posting it here.

I am looking forward to the results of this experimental challenge, both for the mathematical insights gained, and for the lessons learned on how to crowdsource AI assistance in mathematics in a responsible fashion.