Number theory breakthroughs at the dawn of a new era

Like everyone else, I'm trying to make sense of the times we are living through. Will human math exposition (like these notes) be worth anything beyond the prime number year 2027? Can I get a brain chip from Neuralink that lets me read thousands of math papers every week? What does it mean to be human?

As part of this processing, I'd like to simply share some examples of results people have been posting recently on topics close to the RH Saga. Not all of these may have relied on AI! I'll add some comments at the end.

Irrationality of special values

Throughout the RH Saga videos, we studied the Riemann zeta function $\zeta(s)$ and the L-function we called $L_A(s)$, a Dirichlet L-function also denoted $L_{-4}(s)$, associated to the character of conductor $-4$:

\[ L_{-4}(s) = 1 - \frac{1}{3 ^s} + \frac{1}{5 ^s} - \frac{1}{7 ^s} + \frac{1}{9 ^s} \ldots \]

You may recall that the value of this L-function at $s=1$ equals $\frac{\pi}{4}$ (going back to Leibniz, and before him to Madhava of the Kerala School in India). The value at $s=2$, i.e. the series

\[ G = 1 - \frac{1}{3 ^2} + \frac{1}{5 ^2} - \frac{1}{7 ^2} + \frac{1}{9 ^2} \ldots = 0.9159655941772\ldots \]

is known as Catalan's constant. A new preprint of Zhi-Wei Sun of Nanjing University claims to prove that $G$ is irrational. The claim has been challenged by another preprint of Dane Wachs, who claims there are some crucial errors in certain estimates of Sun. Wachs himself has been known to post complete bullshit in the general direction of $\mathbb{F}_1$, as in this derived cohomological framework for the BSD conjecture.

As for the Riemann zeta function, a preprint of Aabir Fauzan (posted on CERN's Open Science server Zenodo) claims to prove that $\zeta(5)$ is irrational! This (if correct) is the first such theorem on the Riemann zeta function since Apéry's famous and surprising proof on $\zeta(3)$ from 1978. Unlike the other authors mentioned in this post, it seems like Fauzan has no previous history of posting any kind of research papers.

Sylvester's Conjecture

We have discussed before the basic question of which primes $p$ can be expressed as $x ^2 + n y ^2$, for some fixed integer $n$. Here $x$ and $y$ are integers, and this question is one of the gateways to the Langlands program.

A similar problem is this: Which primes $p$ can be expressed as $x ^3 + y ^3$, where $x$ and $y$ are now rational numbers? For example, we can write

\[ 17 = (\frac{18}{7}) ^3 + (\frac{-1}{7}) ^3 \]

While the first question is related to the Langlands program, the second is closer to the BSD conjecture, and there is now a proof that in the cases where $p$ is congruent to $4$ or $7$ modulo $9$, such a cube sum representation can be found. The proof is due to Hongbo Yin, and you can read more about Sylvester's conjecture in this survey by Dasgupta and Voight.

Gaps between primes

Open AI (i.e. Kevin Barreto and possibly other mathematicians) recently released their results on short gaps between primes, that there are infinitely many pairs of primes with mutual distance at most 186. In the "opposite" direction, they also posted on improved bounds for long gaps, improving on results by very famous mathematicians like Ford, Green, Konyagin, Tao, and Maynard.

These results were certified in Lean (in the first case modulo certain known theorems related to Deligne's proof of the local Riemann Hypothesis). They were posted as pdfs on the Open AI website and announced on X.

Special values of the quantum dilogarithm

One of the great and classical research programs in number theory is Hilbert's 12th problem, which aims to describe every abelian extension of a given number field in terms of special values of transcendental functions. The simplest case is when the number field is $\mathbb{Q}$, and one can use roots of unity, which are values of the exponential function. The most famous classical case is that of an imaginary quadratic number field, where one can use the modular $j$-function; this is sometimes called Kronecker's Jugendtraum.

Now, a new paper of Radchenko and Wheeler studies a very interesting special function called Faddeev's modular quantum dilogarithm. The motivation is to solve Hilbert's 12th problem in the case of a real quadratic number field, and quite possibly they could achieve this goal in subsequent papers, in a more direct or explicit form than in the earlier breakthrough of Dasgupta and Kakde, who use $p$-adic integration for infinitely many primes $p$.

A brave new world?

One University of Chicago professor (a Bayesian statistician!) miraculously managed to write over 14 books and 200 research papers in less than a year (article here, behind a Washington Post paywall). If you are a regular person trying to follow new research developments, it is genuinely hard to know whether a paper is worth your attention or whether it's AI slop, and there is a serious risk that high-quality papers of not-so-famous authors will now simply drown in the deluge of slop. This problem will surely get much worse in the near future, and the old system of verification by peer review and journal acceptance will likely break down, as the journals themselves are flooded by submissions, and there are simply not enough human reviewers to keep up with the flow. We will be needing new mechanisms for allocating both credit and attention; no-one knows how to do this, although some very early experimental attempts are already underway.

In the paper of Yin (on Sylvester's conjecture), there is no acknowledgement of AI use, and it seems entirely possible that no AI was used.

The pdf on $\zeta(5)$ says: A generative AI tool was used in a supporting role for editing the exposition, proofreading, LATEX preparation, and consistency checks of the arguments, calculations, and references. The author takes full responsibility for the content and correctness of this work.

The paper of Sun contains the following acknowledgement: The author’s many rounds of conversations with AI provide the basis of this paper. At first, the author asked AI to prove the irrationality of Catalan’s constant via the Calegari-Dimitrov-Tang method. Though AI made many attempts but we failed again and again. Later the author realized that we should use suitable weights and introduce weighted tails defined in (1.2). This novel idea made the proof aided by AI finally practical and successful. The needed numerical data in the proof were produced by AI. The whole proof has passed the verification of Chatgpt 5.6 Solar.

The quantum dilogarithm paper has this: We used LLMs exclusively for the purpose of checking the mathematical calculations in Appendix B, which helped identify several mistakes in an earlier draft. All final calculations were independently verified by the authors, who take full responsibility for the results.

Among the many intelligent commentaries made on these developments, I'd like to highlight the posts of Persiflage (e.g. Ways to improve mathematics (an introduction)), Daniel Craven's Where will all the papers go, and Daniel Litt's A beginning for mathematics, all of them with many interesting comments.