Riesz Means of Quadratic Class Numbers

Technical Summary: Riesz Means of Quadratic Class Numbers

Problem Statement
The paper addresses the asymptotic behavior of weighted Riesz means involving Hurwitz class numbers $H(n)$ (associated with negative discriminants) and real quadratic class number traces $Tr_d(1)$ (associated with positive discriminants). While the growth of class numbers for negative discriminants was resolved in the 20th century through the work of Heilbronn, Siegel, and others, the analogous conjectures for positive discriminants remain open. The authors aim to derive a precise asymptotic formula for a weighted average of these quantities, specifically addressing the challenge posed by the nonholomorphic nature of the generating functions for positive discriminant class numbers.

Methodology
The authors employ a framework extending the theory of modular forms to sesquiharmonic Maass forms. The methodology proceeds through the following steps:

1. Definition of L-functions for Sesquiharmonic Forms:
   The paper introduces $L$-functions for weight $1/2$ sesquiharmonic Maass forms of moderate growth. Unlike holomorphic modular forms, sesquiharmonic forms possess Fourier expansions containing three distinct series: a holomorphic part ($c(n)q^n$), a nonholomorphic part involving the incomplete gamma function $\beta(y)$ ($b(n)\beta(-4nv)q^n$), and a third nonholomorphic part involving a special function $\alpha(y)$ ($a(n)\alpha(4nv)q^n$).
   The authors define a completed Dirichlet series $\Lambda_N(F, s)$ that incorporates these three components using special functions $W_{1/2}(s)$ and $Z(s)$, which are defined via integrals involving the incomplete gamma function and the function $\alpha(y)$.

2. Analytic Properties of the L-functions:
   The authors prove that $\Lambda_N(F, s)$ admits a meromorphic continuation to the entire complex plane. They establish a functional equation relating $\Lambda_N(F, s)$ to $\Lambda_N(G, 1/2 - s)$, where $G$ is the image of $F$ under the Fricke involution $W(N)$. The poles of the L-function are explicitly calculated in terms of the constant and logarithmic coefficients of the Fourier expansion of $F$ and $G$.

3. Derivation of a Summation Formula:
   Utilizing Perron's formula and contour integration, the authors derive a general summation formula for the coefficients of sesquiharmonic Maass forms. This formula generalizes previous results for mock modular forms (by Beckwith, Diamantis, Gupta, Rolen, and Thalagoda) and classical results by Chandrasekharan and Narasimhan.
   Crucially, the formula includes additional terms arising from the two non-holomorphic generating series ($\alpha$ and $\beta$) specific to sesquiharmonic forms. These terms manifest as integrals involving Bessel functions and hypergeometric functions.

4. Application to Quadratic Class Numbers:
   The authors apply this general summation formula to a specific sesquiharmonic Maass form $\hat{Z}(\tau)$ introduced by Duke, Imamoğlu, and Tóth. The Fourier coefficients of this form correspond to the Hurwitz class numbers $H(n)$ and the real quadratic traces $Tr_d(1)$. By computing the action of the Fricke involution on $\hat{Z}(\tau)$, they determine the necessary coefficients to plug into the summation formula.

Key Contributions and Results

• Theoretical Extension: The paper successfully defines and analyzes $L$-functions for weight $1/2$ sesquiharmonic Maass forms, a class of functions that had not been treated with this specific $L$-function machinery previously. This includes the proof of the functional equation and the determination of residues.
• Generalized Summation Formula (Theorem 4.1): The authors provide a closed-form summation formula for the Riesz means of coefficients of sesquiharmonic Maass forms. This formula explicitly accounts for the three distinct components of the Fourier expansion, resulting in a more complex structure than those for holomorphic or harmonic Maass forms.
• Asymptotic Formula for Class Numbers (Theorem 1.2): The primary result is an asymptotic formula for a weighted Riesz mean of $Tr_n(1)$ and $H(n)$. For $\rho > 3/2$ and any $\epsilon > 0$, the paper establishes:
  $$\sum_{n \le x} \left(1 - \frac{n}{x}\right)^\rho \left( Tr_n(1) + H(n)\left(\frac{x}{n} - 1\right)^{1/2} \frac{2F_1(\dots)}{\sqrt{2n}(\rho + 1/2)} \right) = \frac{\pi x}{6\sqrt{2}(\rho+1)} + O(x^{1/2+\epsilon})$$
  where $2F_1$ is the Gauss hypergeometric function.
• Numerical Verification: The paper includes a numerical example (Example 1) for $\rho=2$, demonstrating the slow convergence of the sums toward the predicted constant, consistent with the theoretical error term.

Significance
The paper claims significance in bridging the gap between the theory of harmonic Maass forms and the more complex sesquiharmonic Maass forms. By extending the $L$-function methodology and summation formulas to this broader class, the authors provide a tool to analyze the growth of real quadratic class numbers, a problem that has remained open since Gauss. The work generalizes recent results on mock modular forms and offers a concrete asymptotic formula for a weighted average of class numbers, contributing to the understanding of the distribution of these arithmetic invariants. The authors note that their method extends the work of Chandrasekharan and Narasimhan and the recent preprint [BDG+], adapting it to handle the additional non-holomorphic terms inherent in sesquiharmonic forms.