Machine learning the vanishing order of rational $L$-functions
Abstract
In this paper, we apply data-scientific methods to learn the vanishing order of rational $L$-functions for a heterogeneous dataset. Each $L$-function is represented in our data by finitely many Dirichlet coefficients, the normalisation of which depends on the context. We observe murmuration-like patterns in averages across our dataset, find that PCA clusters rational $L$-functions by their vanishing order, and record that LDA and neural networks accurately predict this quantity using a relatively small number of Dirichlet coefficients. In general, there is no rigorous method to compute vanishing order; thus, fast and accurate predictions of these values would be useful in a variety of applications.
Fields of Study & Subjects
Polynomial and algebraic computation
Advanced Numerical Analysis Techniques
Publication details
- Journal / source
- Advances in Theoretical and Mathematical Physics
- Volume
- 29
- Issue
- 8
- Pages
- 2381–2401
- Publication date
- 1 January 2026
- Language
- EN