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Precision Asymptotics for Partitions Featuring False-Indefinite Theta Functions

Kathrin Bringmann, William Craig, Caner Nazaroglu

TL;DR

This paper develops a circle-method framework around false-indefinite theta functions tied to Maass forms to study the asymptotics of Fourier coefficients arising from parity-separated partitions. By leveraging mock Maass theta theory, Mordell-type obstructions, and modular completions, the authors obtain a high-precision asymptotic expansion for the coefficients $\alpha_j(n)$, explicitly capturing all exponentially growing terms and enabling Hardy–Ramanujan–Rademacher-type formulas under suitable conditions. The results demonstrate detailed control over both principal and non-principal contributions via the obstruction representation and circle integration, and they connect combinatorial partition data to the analytic structure of Maass forms. The methods are broad and suggest extensions to other modular-adjacent objects, with potential applications to exact formulas for weights where circle-method techniques yield convergent expansions. A key outcome is the explicit leading-exponential terms for $\alpha_j(n)$, expressed in terms of the local kernel data $\Phi_{\ell,0}$ and its derivatives at $t=0$, anchoring the asymptotics in the local analytic structure of the associated false-indefinite theta functions.

Abstract

Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these relations pertaining to Maass forms and false-indefinite theta functions. In particular, we can systematically distinguish and study the class of false-indefinite theta functions related to Maass forms. A crucial component here is the framework of mock Maass theta functions built by Zwegers in analogy with his earlier work on indefinite theta functions and their application to Ramanujan's mock theta functions. Given this understanding, a natural question is to what extent one can utilize modular properties to investigate the asymptotic behavior of the associated Fourier coefficients, especially in view of their relevance to combinatorial objects. In this paper, we develop the relevant methods to study such a question and show that quite detailed results can be obtained on the asymptotic development, which also enable Hardy-Ramanujan-Rademacher type exact formulas under the right conditions. We develop these techniques by concentrating on a concrete example involving partitions with parts separated by parity and derive an asymptotic expansion that includes all the exponentially growing terms.

Precision Asymptotics for Partitions Featuring False-Indefinite Theta Functions

TL;DR

This paper develops a circle-method framework around false-indefinite theta functions tied to Maass forms to study the asymptotics of Fourier coefficients arising from parity-separated partitions. By leveraging mock Maass theta theory, Mordell-type obstructions, and modular completions, the authors obtain a high-precision asymptotic expansion for the coefficients , explicitly capturing all exponentially growing terms and enabling Hardy–Ramanujan–Rademacher-type formulas under suitable conditions. The results demonstrate detailed control over both principal and non-principal contributions via the obstruction representation and circle integration, and they connect combinatorial partition data to the analytic structure of Maass forms. The methods are broad and suggest extensions to other modular-adjacent objects, with potential applications to exact formulas for weights where circle-method techniques yield convergent expansions. A key outcome is the explicit leading-exponential terms for , expressed in terms of the local kernel data and its derivatives at , anchoring the asymptotics in the local analytic structure of the associated false-indefinite theta functions.

Abstract

Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these relations pertaining to Maass forms and false-indefinite theta functions. In particular, we can systematically distinguish and study the class of false-indefinite theta functions related to Maass forms. A crucial component here is the framework of mock Maass theta functions built by Zwegers in analogy with his earlier work on indefinite theta functions and their application to Ramanujan's mock theta functions. Given this understanding, a natural question is to what extent one can utilize modular properties to investigate the asymptotic behavior of the associated Fourier coefficients, especially in view of their relevance to combinatorial objects. In this paper, we develop the relevant methods to study such a question and show that quite detailed results can be obtained on the asymptotic development, which also enable Hardy-Ramanujan-Rademacher type exact formulas under the right conditions. We develop these techniques by concentrating on a concrete example involving partitions with parts separated by parity and derive an asymptotic expansion that includes all the exponentially growing terms.
Paper Structure (14 sections, 26 theorems, 232 equations, 5 figures, 1 table)

This paper contains 14 sections, 26 theorems, 232 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and Statement of Results
  2. Preliminaries
  3. Maass forms
  4. Mock Maass and false-indefinite theta functions
  5. False-indefinite theta functions and $p_\mathrm{od}^\mathrm{eu}(n)$
  6. The Obstruction to Modularity
  7. Expanding the obstruction to modularity
  8. Mordell type representation of the modular transformation
  9. Bounds on Non-Principal Parts
  10. Proof of Theorem \ref{['T: Main Theorem']}
  11. Proof of Corollary \ref{['cor:leading_exponential_first_few_example']}
  12. Conclusion
  13. Estimates on the Fourier Coefficients of False-indefinite Theta Functions
  14. Taylor Coefficients of the Integral Kernel

Key Result

Theorem 1.1

For $n\in\mathbb N$, $j \in \{0,1\}$, and with $\Delta_0 := - \frac{1}{48}$ and $\Delta_1 := \frac{23}{48}$, we have where the multiplier $\psi_{h,k}(j,\ell)$ is defined in eq:multiplier_system_definition, $\Phi_{\ell,\frac{h'}{k}}$ is given in eq:integral_kernel_definition with $0 \leq h' < k$ chosen to satisfy $h h' \equiv -1 \ \left( \mathrm{mod} \ k \right)$, and $\operatorname{PV} \int

Figures (5)

  • Figure 1:
  • Figure 2:
  • Figure 3: $\mathcal{C}_3$ and $\mathcal{C}_4$ are circular arcs centered at zero and $\mathcal{C} := \mathcal{C}_2 + \mathcal{C}_4 + \mathcal{C}_5 + \mathcal{C}_3 + \mathcal{C}_1$.
  • Figure 4:
  • Figure 5:

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • Proposition 3.1
  • Remark
  • proof : Proof of Proposition \ref{['P: U_j and u_j transformation']}
  • Lemma 4.1
  • ...and 39 more