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
RH is a precise statement, and in one sense what it means is clear, but what it is connected with, what it implies, where it comes from, can be very unobvious. Martin Huxley Previously in the RH Saga, we introduced the "Big Five" problems motivating the idea of
I'm working on writing up Chapter 3 on the Generalized Riemann Hypothesis, but in the meantime, for this weekend, I want to recommend a talk that complements the RH Saga E2 and the associated Chapter 2, and is delivered in much of the same spirit. It is a
A number of deep problems about prime numbers emerge from the simple idea of taking a polynomial and trying to understand which primes appear among its values, and in particular whether there is a finite or an infinite number. We speak then of primes of the form $n^2+1$
Every four years, researchers from around the world gather at the International Congress of Mathematicians. Fields medals are awarded, and many lectures are given by the greatest experts in the world, aimed at a general audience of mathematicians. A lecture at the ICM (typically given both as a live talk
The "Himalayan peaks" that hold the secrets of these nonabelian reciprocity laws challenge humanity, and, with the visionary Langlands program, we have mapped out before us one means of ascent to those lofty peaks. The recent work of Wiles suggests that an important case (the semistable case) of
Yet another really nice talk hosted by the Clay Mathematics Institute is this talk from 2025 by Kannan Soundararajan on progress in the theory of zeta and L-functions motivated by the Riemann Hypothesis. It is in some ways even more accessible than the two earlier talks by Sarnak, but
I'm in the process of preparing next week's Chapter 2, on the Langlands program and the landscape of different kinds of L-functions. Most research talks on the Langlands program are incomprehensible unless you already have huge amounts of technical background. But I have found one
The logarithmic integral function $\mathrm{li}(x)$ is defined as \[ \mathrm{li}(x) = \int_0^{x} \frac{\mathrm{d}t}{\log(t)} \] where $\log$ is is the natural logarithm, and the integral is understood as the Cauchy principal value at $t=1$. The function $\mathrm{li}(x)$ is for $x&
The Riemann Hypothesis may be the greatest unsolved math problem of all time. In his book Zeta and L-functions of varieties and motives, Bruno Kahn calls it an unapproachable mystery. In a short YouTube clip, Terence Tao likened it to climbing a sheer cliff face, a mile high, with
For obvious reasons, it is hard to find very accessible introductions to L-functions. Here are three pieces of exposition where such an introduction is attempted. Perhaps you will find them interesting! And perhaps you can send them to your uncle in order to explain why you love L-functions.
Part of the philosophy I hope to promote at PeakMath is to play! Play around with the mathematical entities you encounter! Try silly things, make fun observations, and don't worry too much about whether your findings are "important" or not. In an interesting philosophical paper, Alexandra
In June 2018 a conference was held in Bristol called Perspectives on the Riemann Hypothesis. The talks from this conference are not on YouTube, but can be found on the conference webpage. Many of these talks are excellent, but today I specifically want to recommend the talk by Sarnak, called
On this very day, May 6th, exactly 5! years have passed since the birth of André Weil. One of the truly great mathematical minds of the 20th century, Weil proposed his famous Weil Conjectures on the structural properties of local zeta functions. They were eventually resolved by himself in the