Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Summation formulas for Hurwitz class numbers and other mock modular coefficients

Olivia Beckwith, Nikolaos Diamantis, Rajat Gupta, Larry Rolen, Kalani Thalagoda

TL;DR

The paper develops a novel framework of summation formulas for mock modular forms of polynomial growth, integrating Chandrasekharan–Narasimhan ideas with harmonic Maass form tools. It constructs $L$-functions for harmonic Maass forms, establishes a main summation formula linking coefficient sums to dual mock-modular data via Bessel-type kernels, and derives asymptotics for sums of coefficients such as Hurwitz class numbers and coefficients of negative half-integer weight Eisenstein series. The results are refined to extend the admissible range of parameters (notably $k= frac{3}{2}$) and are complemented by numerical checks, illustrating the arithmetic power of mock-modular summation formulas. A Converse Theorem is also provided, showing that these summation identities force a functional equation and hence modularity of the underlying sequences, highlighting a deep connection between mock modular phenomena and classical automorphic structure.

Abstract

We prove a formula for weighted sums of the first $n$ coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of certain quadratic Dirichlet $L$-values. Our formula is a mock modular version of a Bessel-sum identity proved by Chandrasekharan and Narasimhan for Dirichlet series satisfying a functional equation. Our proof utilizes $L$-functions for mock modular Eisenstein series defined by Shankadhar and Singh.

Summation formulas for Hurwitz class numbers and other mock modular coefficients

TL;DR

The paper develops a novel framework of summation formulas for mock modular forms of polynomial growth, integrating Chandrasekharan–Narasimhan ideas with harmonic Maass form tools. It constructs -functions for harmonic Maass forms, establishes a main summation formula linking coefficient sums to dual mock-modular data via Bessel-type kernels, and derives asymptotics for sums of coefficients such as Hurwitz class numbers and coefficients of negative half-integer weight Eisenstein series. The results are refined to extend the admissible range of parameters (notably ) and are complemented by numerical checks, illustrating the arithmetic power of mock-modular summation formulas. A Converse Theorem is also provided, showing that these summation identities force a functional equation and hence modularity of the underlying sequences, highlighting a deep connection between mock modular phenomena and classical automorphic structure.

Abstract

We prove a formula for weighted sums of the first coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of certain quadratic Dirichlet -values. Our formula is a mock modular version of a Bessel-sum identity proved by Chandrasekharan and Narasimhan for Dirichlet series satisfying a functional equation. Our proof utilizes -functions for mock modular Eisenstein series defined by Shankadhar and Singh.
Paper Structure (19 sections, 36 theorems, 155 equations, 3 figures)

This paper contains 19 sections, 36 theorems, 155 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivating Example: Class Numbers
  3. Summation formulas for mock modular forms
  4. Analytic preliminaries and some special functions
  5. Preliminaries
  6. Special function arising from the non-holomorphic part
  7. Polynomial growth harmonic Maass forms
  8. Notation
  9. Harmonic Maass forms of polynomial growth
  10. $L$-functions for polynomial growth harmonic Maass forms
  11. A summation formula for polynomial growth harmonic Maass forms
  12. Asymptotic formula
  13. Refining Theorem \ref{['thm:summationformula']}
  14. Extending the range for $\rho$
  15. Applications
  16. ...and 4 more sections

Key Result

Theorem 1

Let $H(n)$ denote the Hurwitz class number. For $\rho>1$ and $\epsilon > 0$, we have

Figures (3)

  • Figure 1: Ratio of $\sum_{n\leq x} H(n)(x-n)^{\rho}$ and the leading term $\frac{ \pi^{\frac{3}{2}}\Gamma(\rho+1)}{24 \Gamma \left (\rho + \frac{5}{2} \right )}x^{\rho + 3/2}$ for values of $\rho = 0.5 + 10^{-5},1,1.5,2$
  • Figure 2: The ratio of $\sum_{n\leq x} H(n)(x-n)^{\rho} - \frac{ \pi^{\frac{3}{2}}\Gamma(\rho+1)}{24 \Gamma \left (\rho + \frac{5}{2} \right )}x^{\rho + 3/2}$ and the first error term $\frac{x^{\rho + 1}3(1+i)i^{3/2}}{16\Gamma(\rho + 2)}$
  • Figure 3: $\sum_{n\leq x} H(n)(x-n)^{\rho}$ and the potential leading term $\frac{ \pi^{\frac{3}{2}}\Gamma(\rho+1)}{24 \Gamma \left (\rho + \frac{5}{2} \right )}x^{\rho + 3/2}$ for values of $\rho = 0,-0.1,-0.5$

Theorems & Definitions (64)

  • Theorem
  • Theorem
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • ...and 54 more