Murmurations: a case study in AI-assisted mathematics

TL;DR

This paper introduces 'murmurations', a novel phenomenon in arithmetic discovered through AI analysis, which reveals subtle number-theoretic information and connects to major conjectures and theories in number theory.

Contribution

It presents the discovery and analysis of murmurations, a new phenomenon in arithmetic, using machine learning interpretability tools, bridging AI and number theory.

Findings

Murmurations encode information about Frobenius traces.

They relate to the Birch and Swinnerton-Dyer conjecture.

Connect to random matrix theory in number theory.

Abstract

We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard interpretability tools from machine learning, including principal component weightings, saliency curves, and convolutional filters. Although discovered computationally, they constitute a genuinely new and intriguing phenomenon in arithmetic that can be formulated and investigated using established tools of number theory. In particular, murmurations encode subtle information about Frobenius traces and naturally belong to the framework of arithmetic statistics. More precisely, murmurations connect to central themes surrounding the conjecture of Birch and Swinnerton-Dyer and perspectives from random matrix theory. In this paper, we present an overview of murmurations, contextualizing them within number theory and AI.