Predicting root numbers with neural networks

Abstract

In this paper, we report on two machine learning experiments in search of statistical relationships between Dirichlet coefficients and root numbers or analytic ranks of certain low-degree [Formula: see text]-functions. The first experiment is to construct interpretable models based on murmurations, a recently discovered correlation between Dirichlet coefficients and root numbers. We show experimentally that these models achieve high accuracy by learning a combination of Mestre–Nagao-type heuristics and murmurations, noting that the relative importance of these features varies with degree. The second experiment is to search for a low-complexity statistic of Dirichlet coefficients that can be used to predict root numbers in polynomial time. We give experimental evidence and provide heuristics that suggest this cannot be done with standard machine learning techniques.