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,
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
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,
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
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-
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
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(