Something quite remarkable happened yesterday. Over the last few months a number of impressive new results were found by AI models (notably the counter-examples to the Jacobian Conjecture in 3D and to the Unit Distance Conjecture of Erdos). You can read about the latter one in Quanta or read
In the next chapter of the RH Saga, I want to discuss operations on L-functions, and say something meaningful about the relationship between these operations and a possible future theory of $\mathbb{F}_1$. One of the most important perspectives on L-function operations comes from the Principle of
In Enrico Bombieri's official Millennium problem description of RH, he says the following: It is outside the scope of this article even to outline the definition of global L-functions, referring instead to Iwaniec and Sarnak [IS] for a survey of the expected properties satisfied by them; it
With the kind permission of Elie Studnia, I want to share here two open research problems presented by him at the ANTS "rump session" (i.e. the ultra-short talks). The first one might be for you if you are looking for a concrete computational unsolved problem where
Writing up Chapter 4 of the RH Saga is taking a bit longer than expected, as I'm travelling while also enduring the quite unusual European heatwave. But there is one more important video I can share now to which I want to refer in Chapter 4, and this
From the first day of ANTS, one highlight I'd like to share is the talk on Drinfeld modules by Antoine Leudière, reporting on joint work with Renate Scheidler. In the context of Weil's Rosetta Stone, there is a specific analogy between elliptic curves (in the number
Last week I shared an introductory talk by Ram Murty aiming to answer the question "What is an L-function?". Today I'm driving from Paris to Groningen, and on the way (from a McDonalds near Antwerp) I just want to share one other talk in the
One the best conference series in computational number theory (including many L-function-related themes) is the biannual ANTS series. The 2024 edition was held at MIT, and next week ANTS XVII is happening in Groningen. Personally, I haven't been to a real math conference for years, but
The next chapter of the RH Saga will aim to answer the question "What is an L-function?". As preparation for that chapter, perhaps you will enjoy this beautiful and well-paced talk by Ram Murty covering different notions of L-functions in a panoramic and historical overview,
If you have followed the RH Saga, you have almost certainly also seen many of the other RH-related videos from other YouTube channels. Still, I have collected here five of the best general introductions to RH, in case there are some you want to watch or rewatch. At the
Gerd Faltings was recently awarded the Abel Prize, and there is now a number of nice video lectures explaining his work to more or less specialized audiences. I have chosen here three talks in increasing level of specialization, hoping that at least one of them will provide an interesting watch
One of the most important theorems of elementary number theory is Fermat's little theorem. It says that whenever an integer $a$ is not divisible by a prime $p$, the number $a^{p-1}-1$ is divisible by $p$. In terms of congruences we can write this statement as