PeakMath

It was an audacious proposal.

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.

There are no Siegel zeros.

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.

The talks from Jean-Pierre Serre's 100th birthday conference are out!

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.

Do you love maths and video editing?

Continuing our review of operations on L-functions, this shorter post will cover the key ideas around the Selberg inner product.

Like everyone else, I'm trying to make sense of the times we are living through.

Looking ahead, Chapter 6 of the RH Saga will cover invariants of L-functions.

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 $L_1 \otimes L_2$.

... and what this means for the PeakMath essay competition.

At the phenomenological level this is perhaps the most striking discovery about zeta since Riemann.

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.

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.

By a somber winter night in 1949, André Weil, reading dusty manuscripts of C.

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.

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…

Something quite remarkable happened yesterday.

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 $\mathbb{F}_1$.

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 expec…

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).

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.

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.

Last week I shared an introductory talk by Ram Murty aiming to answer the question "What is an L-function?".

One the best conference series in computational number theory (including many L-function-related themes) is the biannual ANTS series.