Murmurations: a case study in AI-assisted mathematics
This work presents a novel discovery in mathematics using AI tools, potentially advancing number theory research.
The paper introduces 'murmurations', a new phenomenon in arithmetic discovered through AI analysis of large datasets, which connects to number theory concepts like Frobenius traces and the Birch and Swinnerton-Dyer conjecture.
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.