Moments of ideal class counting functions

TL;DR

This paper studies the statistical properties of ideal class counting functions in number fields, connecting them to automorphic forms on GL(d), and computes their moments including cuspidal projections.

Contribution

It introduces the computation of moments for ideal class counting functions and explores their cuspidal projections, advancing understanding of their automorphic form connections.

Findings

Computed moments of ideal class counting functions

Analyzed moments of cuspidal projections

Linked counting functions to automorphic form coefficients

Abstract

We consider the counting function of ideals in a given ideal class of a number field of degree $d$. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on $\text{GL}(d)$, typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.