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
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
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
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
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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(
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
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
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
... 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
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
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
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.
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
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
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
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
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