Lectures on the quasi-Riemann hypothesis

Two excellent talks have now appeared that explain the proof of the new quasi-RH theorem. The talk of Maksym Radziwill is somewhat more accessible, while the talk of Valentin Blomer is a bit more technical.

Weekend research talk: Julia Stadlmann on bounded gaps

I found this really nice talk by Julia Stadlmann, on her recent improvements on bounded gaps between primes, more precisely the result that there is an infinite number of prime pairs with distance at most 240. Almost immediately after her preprint was posted, several AI companies posted even better bounds,

RH Saga Chapter 6: Invariants of L-functions (intro)

The world of L-function invariants is mindblowingly and extraordinarily rich. In this chapter, I want to make an attempt at organising and structuring this world. By an invariant, I will mean a rule which attaches a number to an L-function. In other words, it is a machine which

On Ramanujan and the history of modular forms

For this weekend, I'd like to recommend a gem of a lecture by Ram Murty, on the mathematical life of Ramanujan and the history of modular forms. This very down-to-earth talk covers among other things Ramanujan's work on partitions, the tau function, Maass forms,

Human-readable papers on Siegel zeros

Ben Green has posted a new paper on Hexagon, with a five-page proof of the groundbreaking OpenAI result on Landau-Siegel zeros. That this kind of short and simple proof exists for such a deep statement raises many questions about the nature of human creativity and the structure and

RH Saga Chapter 5.5: A Tannakian category?

It was an audacious proposal. As far as I know, nothing of the sort had ever been imagined before. (James Arthur, describing the Tannakian proposal of Langlands made around 1977) We are now at the heart of the search for $\mathbb{F}_1$. We can say quite precisely what L-

A transcript of Sarnak's latest talk

In exploring various ways to leverage these new AI capabilities for human understanding, one small experiment I did was to transcribe Sarnak's wonderful lecture from the Serre birthday conference. Watching videos takes a lot of time, so in case you find it valuable, here is the transcript, with

The Time of the Orc has Come

There are no Siegel zeros. There are no zeros of the Riemann zeta function with real part bigger than $\frac{7}{8}$. There are infinitely many primes for which $2$ is a primitive root. These three statements are special cases of the somewhat more general results found by OpenAI and

RH Saga Chapter 5.4: Perturbations

There are many situations in the literature in which two Dirichlet series arise which have identical Euler products except at a finite number of primes, where the Euler factors differ. Motivated by all of these examples, I want to define a perturbation of an Euler product as any operation that

New talk of Sarnak for Serre's 100th birthday!

The talks from Jean-Pierre Serre's 100th birthday conference are out! Here is the conference program. Below is the talk by Sarnak, where you will see yet another explanation of many ideas from the RH Saga, like complexity and subconvexity, sometimes with some new twists you might not

What makes for a really good mathematical essay?

I added some more information over at the Essay competition page, including some examples of really good essays and some notes on what makes an essay good. Check it out and spread the word in case you have students or friends who might be interested in submitting something! Below is

PeakMath seeks video editor

Do you love maths and video editing? Would you be interested in working with us at PeakMath on new YouTube videos? Read more here and get in touch via the contact form!

RH Saga Chapter 5.3: Selberg inner product

Continuing our review of operations on L-functions, this shorter post will cover the key ideas around the Selberg inner product. You may recall from the end of Episode 5 that given two L-functions \[ L_1(s) = \sum \frac{a_n}{n ^s} \quad \textrm{and} \quad L_2(

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

Weekend research talk: Moments of L-functions

Looking ahead, Chapter 6 of the RH Saga will cover invariants of L-functions. One topic omitted in the Episode 6 video is the notion of moments. This term covers an important class of invariants attached either to an individual L-function, or to a family of L-functions. They

RH Saga Chapter 5.2: Lambda-ring operations

Given two L-functions $L_1$ and $L_2$, we can form their direct sum $L_1 \oplus L_2$, as well as their tensor product $L_1 \otimes L_2$. These two fundamental operations of addition and multiplication make the set $\mathbb{L}$ of all L-functions into a

A Millennium problem has fallen

... and what this means for the PeakMath essay competition. The entire internet (or at least my mathematical bubble) seems to be in shock at the news. OpenAI has released a solution to the Navier-Stokes problem. For a brief summary of events over the past few days and the controversy

RH Saga Chapter 5.1: Families of L-functions

At the phenomenological level this is perhaps the most striking discovery about zeta since Riemann. Let's keep going in our quest for clues for a possible theory of the field with one element! Last time, we emphasized the importance of viewing L-functions not as isolated objects but

Weekend research talk: Taylor Dupuy on Deninger's RH proof strategy

One of the dreams related to the Generalized Riemann Hypothesis is that the zeros of an L-function might have a spectral interpretation, i.e. they might be eigenvalues of some operator, or at least closely related to such eigenvalues. Since there are infinitely many nontrivial zeros, the operator seemingly

Weekend research talk: Random matrix theory and zeta functions

As a warmup to the next chapter of the RH Saga, here is a talk by Peter Sarnak from 2014, in which he gives a beautiful survey of the history of random matrix models for zeros of L-functions, and an overview of the theory of families of L-functions.

The latest talk of Alain Connes on absolute geometry

Among the many different approaches to $\mathbb{F}_1$ proposed in the literature, one of the most serious is the long-running program of Alain Connes, building on ideas developed with Consani and other collaborators over several decades. I received this video in a message from René Slabbekoorn and wanted

RH Saga Chapter 5: Operations on L-functions (intro)

By a somber winter night in 1949, André Weil, reading dusty manuscripts of C.-F. Gauss, discovers the existence of a spectacular treasure lying somewhere in the deep jungles of arithmetic geometry. He manages to sketch a map to reach it but is unable to mount an expedition to embark

Weekend research talk: Barry Mazur on Bridges between Geometry and Number Theory

Note: The previous post (on a possible direction for a competition essay) did not go out by email to everyone, so check it out if you're interested. Now, in this very friendly and accessible talk by Barry Mazur, he talks about some of the themes we have touched

Possible essay topic: Generalized Bernoulli numbers

If you're interested in writing up an essay for the PeakMath Essay Competition and you're looking for something cool to write about, one suggestion is to learn about Bernoulli numbers, Bernoulli polynomials, and generalized Bernoulli numbers, and write up an exposition of some research paper based