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Moments of ideal class counting functions

Kam Cheong Au

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.

Moments of ideal class counting functions

Abstract

We consider the counting function of ideals in a given ideal class of a number field of degree . This describes, at least conjecturally, the Fourier coefficients of an automorphic form on , typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.
Paper Structure (8 sections, 21 theorems, 141 equations, 1 table)

This paper contains 8 sections, 21 theorems, 141 equations, 1 table.

Table of Contents

  1. Introduction
  2. Full partial zeta series
  3. General considerations
  4. Ideal class counting function
  5. Cuspidal partial zeta series
  6. Criterion of being cuspdial
  7. Cuspidal version of $a(\sigma,n)$
  8. Epistein zeta function of non-fundamental discriminant

Key Result

Lemma 2.1

We have $L(\chi,s) = L(\widetilde{\chi},s) = L(\widetilde{\chi}^\text{ind},s)$, that is

Theorems & Definitions (59)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 49 more