# Murmurations: a case study in AI-assisted mathematics

arXiv: https://arxiv.org/abs/2603.09680 · HTML: https://mlsanity.com/paper/2603.09680/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 53 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Builds on (most cited first, 19 of 19)

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- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
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- [Machine-Learning Mathematical Structures](https://mlsanity.com/paper/2101.06317/) (2021)
- [Deep-Learning the Landscape](https://mlsanity.com/paper/1706.02714/) (2017)
- [The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning](https://mlsanity.com/paper/1812.02893/) (2018)
- [Learning Algebraic Structures: Preliminary Investigations](https://mlsanity.com/paper/1905.02263/) (2019)
- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [Is deep learning a useful tool for the pure mathematician?](https://mlsanity.com/paper/2304.12602/) (2023)
- [Graph Laplacians, Riemannian Manifolds and their Machine-Learning](https://mlsanity.com/paper/2006.16619/) (2020)
- [Machine-Learning Arithmetic Curves](https://mlsanity.com/paper/2012.04084/) (2020)
- [Machine-Learning Number Fields](https://mlsanity.com/paper/2011.08958/) (2020)
- [Machine learning detects terminal singularities](https://mlsanity.com/paper/2310.20458/) (2023)
- [Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer](https://mlsanity.com/paper/1911.02008/) (2019)
- [Mathematical conjecture generation using machine intelligence](https://mlsanity.com/paper/2306.07277/) (2023)
- [Machine learning the dimension of a Fano variety](https://mlsanity.com/paper/2309.05473/) (2023)
- [Machine Learning Class Numbers of Real Quadratic Fields](https://mlsanity.com/paper/2209.09283/) (2022)
- [Machine-Learning Kronecker Coefficients](https://mlsanity.com/paper/2306.04734/) (2023)

## Abstract (arXiv)

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.