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Self-Correlations of Hurwitz Class Numbers

Alexander Walker

TL;DR

This work studies self-correlations of Hurwitz class numbers by recasting the shifted convolution sum into a Petersson inner product against a Poincaré series and implementing automorphic regularization to handle non-$L^2$ growth. It establishes a meromorphic continuation and explicit rightmost poles of the associated Dirichlet series $D_\ell(s)$ via a spectral expansion that blends discrete and continuous spectra, with a sharp main-term derived from residues at $s=\tfrac{3}{2}$ and $s=1$. A key technical advance is a rigorous bound for triple inner products of the form $\langle y^k|f|^2, \mu_j\rangle$ when $f$ is a harmonic Maass form of polynomial growth, adapted from Jutila’s Rankin–Selberg method to half-integral weights and non-cuspidal cases. Combining these ingredients with a truncated Perron formula yields a uniform asymptotic for $\sum_{n\le X} H(n)H(n+\ell)$ with a main term proportional to $X^2$ and a power-saving error, and it also clarifies how large-additive-shift behavior is governed by divisor-type sums of $\ell$, while conjecturing a secondary term and smoother variants for refined accuracy.

Abstract

The asymptotic study of class numbers of binary quadratic forms is a foundational problem in arithmetic statistics. Here, we investigate finer statistics of class numbers by studying their self-correlations under additive shifts. Specifically, we produce uniform asymptotics for the shifted convolution sum $\sum_{n < X} H(n) H(n+\ell)$ for fixed $\ell \in \mathbb{Z}$, in which $H(n)$ denotes the Hurwitz class number.

Self-Correlations of Hurwitz Class Numbers

TL;DR

This work studies self-correlations of Hurwitz class numbers by recasting the shifted convolution sum into a Petersson inner product against a Poincaré series and implementing automorphic regularization to handle non- growth. It establishes a meromorphic continuation and explicit rightmost poles of the associated Dirichlet series via a spectral expansion that blends discrete and continuous spectra, with a sharp main-term derived from residues at and . A key technical advance is a rigorous bound for triple inner products of the form when is a harmonic Maass form of polynomial growth, adapted from Jutila’s Rankin–Selberg method to half-integral weights and non-cuspidal cases. Combining these ingredients with a truncated Perron formula yields a uniform asymptotic for with a main term proportional to and a power-saving error, and it also clarifies how large-additive-shift behavior is governed by divisor-type sums of , while conjecturing a secondary term and smoother variants for refined accuracy.

Abstract

The asymptotic study of class numbers of binary quadratic forms is a foundational problem in arithmetic statistics. Here, we investigate finer statistics of class numbers by studying their self-correlations under additive shifts. Specifically, we produce uniform asymptotics for the shifted convolution sum for fixed , in which denotes the Hurwitz class number.
Paper Structure (22 sections, 25 theorems, 150 equations)

This paper contains 22 sections, 25 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Harmonic Weak Maass Forms and Mock Modular Forms
  3. The Shadow Operator xi
  4. Harmonic Maass Forms of Polynomial Growth
  5. Shifted Convolutions via Inner Products
  6. Application to H(z)
  7. Automorphic Regularization
  8. Automorphic Regularization of H(z)
  9. Inner Products Involving Regularization Terms
  10. Spectral Expansion and Rightmost Poles
  11. Classifying the Rightmost Poles of Dh(s)
  12. Bounding Dh(s) in Vertical Strips
  13. Non-Cuspidal Spectral Inner Products
  14. Jutila's Extension of the Rankin--Selberg Method
  15. Representations and Estimates for phi-plus, phi-cross, and phi-minus
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\sigma_\nu(m) = \sum_{d \mid m} d^\nu$ denote the sum-of-divisors function, with the convention that $\sigma_\nu(m) = 0$ for $m \notin \mathbb{Z}$. Fix $\ell \geq 1$ and let $\ell_o$ denote the odd part of $\ell$. Then, for all $\epsilon > 0$, we have

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary
  • Theorem 1.2
  • Lemma 2.1: cf. Theorem 5.2, Muller95
  • Proposition 4.1
  • proof
  • Remark 5.1
  • Proposition 5.2
  • Corollary 5.3
  • proof
  • ...and 38 more