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.

翻译：我们报告了算术中出现的一种引人注目的新现象，我们称之为低语现象。最初通过大型算术数据集的平均实验观测到，低语现象可以使用机器学习中的标准可解释性工具进行检测和分析，包括主成分权重、显著性曲线和卷积滤波器。尽管是通过计算发现的，它们构成了算术中真正新颖且引人入胜的现象，可以使用数论的既定工具进行表述和研究。具体而言，低语现象编码了关于弗罗贝尼乌斯迹的微妙信息，并自然地属于算术统计学的框架。更准确地说，低语现象与围绕伯奇和斯温纳顿-戴尔猜想的核心主题以及随机矩阵理论的视角相关联。在本文中，我们概述了低语现象，将其置于数论和人工智能的背景下进行语境化分析。