Mock Maass Theta Functions

Updated 14 January 2026

Mock Maass theta functions are real-analytic modular objects that decompose into a holomorphic part and a nonholomorphic shadow, obeying explicit transformation laws and differential annihilation.
They are constructed via methods such as Appell–Lerch sums, theta-integral approaches, and Poincaré/Maass Eisenstein lifts, revealing deep arithmetic in their Fourier coefficients.
Their robust modular properties and connections to quantum modularity, automorphic representations, and classical theta series make them central to modern research in number theory.

Mock Maass theta functions are real-analytic modular objects central to the theory of mock modular forms, harmonic weak Maass forms, and their connection to classical theta series. This class of functions unifies analytic, algebraic, and geometric approaches to modularity and is characterized by explicit transformation laws, differential annihilation, and deep arithmetic in their Fourier expansions. Initially motivated by Ramanujan’s enigmatic mock theta functions, recent decades have witnessed a precise classification and rich generalizations across half-integral and integral weights.

1. Foundational Definitions: Harmonic Maass Forms, Shadows, and Completions

A mock Maass theta function is a real-analytic function $H(\tau)$ defined on the upper half-plane (and, in some constructions, continued across the real axis) with the following structure:

Decomposition: $H(\tau) = H^+(\tau) + H^-(\tau)$ , where $H^+$ is holomorphic (“mock theta part”) and $H^-$ is nonholomorphic (“shadow part”) (Kac et al., 2015, Pierre, 2012, Andersen et al., 2016).
Transformation law: $H$ transforms with a prescribed weight $k$ under congruence subgroups or via the Weil representation, often half-integral.
Annihilation: $H$ is annihilated by the weight- $k$ Laplacian (or higher Maass-Jacobi differential operators).
Growth: $H$ is meromorphic or has moderate growth at cusps, with at most simple poles (Kac et al., 2015).
Shadow: The nonholomorphic part $H^-$ can be written in terms of period integrals or incomplete gamma functions involving true modular forms (often unary theta functions) (Li et al., 2013, Andersen et al., 2022).

Half-Integral Weight Example: For instance, Ramanujan’s third-order mock theta function $f(q)$ is completed to a harmonic weak Maass form of weight $1/2$ as

$\widehat f(\tau) = f(q) + \frac{i}{2}\int_{-\bar\tau}^{i\infty} \frac{\Theta_{1,2}(w)}{\sqrt{-i(\tau+w)}}\, dw,$

transforming like a weight $1/2$ modular form (Kac et al., 2015).

2. Analytic and Modular Properties; Jacobi and Weil Representations

Mock Maass theta functions are distinguished by stringent analytic and transformation criteria:

Elliptic properties: Rank-$1$ Jacobi mock theta functions $\mu_{m,s}(\tau, z)$ and their completions $\widehat\mu_{m,s}(\tau,z)$ satisfy controlled elliptic shifts:

$\widehat\mu_{m,s}(\tau, z+1) = e^{2\pi i s}\, \widehat\mu_{m,s}(\tau, z), \qquad \widehat\mu_{m,s}(\tau, z+\tau) = q^{-m} e^{-2\pi i m z} \widehat\mu_{m,s}(\tau, z)$

(Kac et al., 2015).

Modular transformation: For $\tau\mapsto\tau+1$ or $S: \tau\mapsto-1/\tau$ , $z\mapsto z/\tau$ , explicit multiplier structures induce vector-valued Jacobi forms of weight $1$ indexed by $s \pmod{2m}$ (Kac et al., 2015, Andersen et al., 2022).
Weil representation: Many completed mock theta functions transform via the Weil representation on finite quadratic modules; e.g., the family of functions $H_c(z)$ for $c$ , $(c,6)=1$ yield vector-valued weight-$1/2$ harmonic Maass forms (Andersen et al., 2022).

The Kac–Wakimoto characterization: the space of real-analytic functions $F(\tau, z)$ on $\mathbb{H}\times\mathbb{C}$ of weight $1$ and index $m$ satisfying the above elliptic, modular, Maass, pole, and growth conditions is $2m$-dimensional, spanned by the basis $\{ \widehat\mu_{m,s} \}_{s \pmod{2m}}$ (Kac et al., 2015).

3. Construction Techniques and Explicit Examples

Mock Maass theta functions can be explicitly constructed via several methods:

Appell–Lerch sums and nonholomorphic completions: The classic rank-1 construction $\mu_{m,s}(\tau, z)$ plus Zwegers’s real-analytic correction $R_{m,s}(\tau, z)$ produces $\widehat\mu_{m,s}$ with deep modularity and analytic properties (Kac et al., 2015).
Theta-integral approach: Integrals of Siegel theta kernels $\theta_{L+h}(\tau, t)$ over parameter $t$ between negative lines $c_1$ and $c_2$ yield new mock Maass theta functions whose Fourier coefficients are logarithms of algebraic units in real quadratic fields (Li et al., 2023).
Poincaré/Maass Eisenstein lifts: Nonholomorphic Eisenstein series of half-integral weight can be analytically continued and linearly combined to produce harmonic weak Maass forms $F_k$ whose shadow is $\Theta^k$ for $k\in\{3,5,7\}$ (Bhand et al., 2020, Rhoades et al., 2011).
Holomorphic projection and tensor products: Projection of products $F^- G$ where $F$ is Maass, $G$ is modular yields finite divisor-sum recursions for Fourier coefficients of mock theta functions (Imamoglu et al., 2013).
Renormalization and quantum modular forms: The “renormalization” procedure recovers the missing half of the Maass form Fourier coefficients and completes mock theta functions within Don Zagier's quantum modular framework (Li et al., 2013).

Table: Canonical Mock Maass Theta Functions and Their Construction

Family	Construction Method	Transformation/Shadow
$\widehat\mu_{m,s}$	Appell–Lerch + correction (Kac et al., 2015)	Jacobi/Weil rep; index $m$
$F_k$ for $\Theta^k$	Maass Eisenstein lift (Bhand et al., 2020)	Shadow $\Theta^{k}$ , weight $2-k/2$
Theta-integral forms	Siegel kernel integration (Li et al., 2023)	Logarithms of units, weight 0
$H^{(6),1}$	Vector-valued harmonic Maass (Klein et al., 2019)	Shadow theta vector, weight $1/2$
$H_c(z)$	Weil rep. vector (Andersen et al., 2022)	Period integrals, weight $1/2$
Mock-Poincaré series	Contour-integral extension (Andersen et al., 2016)	Dual Poincaré on lower half-plane

4. Fourier Expansions, Trace Formulas, and Arithmetic Data

Mock Maass theta functions often reveal arithmetic content in their Fourier expansions:

Holomorphic/nonholomorphic split: The decomposition $H^+(\tau)+H^-(\tau)$ isolates the mock theta part (coefficients $a(n)$ , q-series) from the nonholomorphic shadow ( $\Gamma$ - or $K$ -Bessel terms integrated against theta kernels or period integrals) (Kac et al., 2015, Bhand et al., 2020).
Trace formulas: Algebraic formulas, especially for Ramanujan's functions $f(q)$ and $\omega(q)$ , express Fourier coefficients as traces of CM values of weakly holomorphic modular functions via Millson or Kudla-Millson theta lifts:

$a_f(n) = \frac{i}{2\sqrt{24n-1}} \left( \operatorname{tr}_F^+(1-24n,1) - \operatorname{tr}_F^-(1-24n,1) \right)$

(Bruinier et al., 2016, Klein et al., 2019).

Arithmetic of units: In weight zero, Fourier coefficients of the harmonic part are logarithms of algebraic units in real quadratic fields, producing deep connections between modular analysis and arithmetic (Li et al., 2023, Rhoades et al., 2011).

5. Differential Operators, Maass-Jacobi Structure, and Shadows

An invariant feature is annihilation by higher-order differential operators:

Maass-Jacobi operator: In the Jacobi setting, the operator

$\Delta_m = \left( \frac{\partial}{\partial \tau} + \frac{i}{4\pi m} \frac{\partial^2}{\partial z^2} \right) \circ \left( \frac{\partial}{\partial \bar\tau} - \frac{i}{4\pi m} \frac{\partial^2}{\partial \bar z^2} \right)$

annihilates the completed mock theta function $\widehat\mu_{m,s}$ (Kac et al., 2015).

$\xi$ -operator and shadow: The anti-holomorphic differential operator $\xi_k(f)(\tau) = 2i v^k \overline{\frac{\partial}{\partial \bar\tau} f(\tau)}$ sends the Maass form $f$ to its shadow modular form, which is often a unary theta series (Pierre, 2012, Bhand et al., 2020).
Lift relations: Connections between Shintani and Millson lifts further relate shadows and mock Maass forms within the framework of Weil representations and modular kernels (Alfes-Neumann et al., 2017).

6. Classification, Dimension, and Families

The explicit classification of mock Maass theta functions is well-developed:

Kac–Wakimoto theorem: The space of modified mock theta functions satisfying fixed analytic, pole, modular, elliptic, Maass, and growth conditions is finite-dimensional—specifically, for Jacobi index $m$ , it is $2m$-dimensional with a basis $\{\widehat\mu_{m,s}\}$ (Kac et al., 2015).
Vector-valued generalizations: Infinite families of vector-valued mock theta functions have been systematically constructed and classified under Weil-type modular representations (Andersen et al., 2022).
Hecke eigenforms: Certain mock Maass theta functions are explicit Hecke eigenforms under operators $T(p^2)$ , paralleling the classical theory (Bhand et al., 2020).

7. Connections, Applications, and Open Directions

Mock Maass theta functions underlie broad areas of arithmetic geometry, combinatorics, and representation theory:

Class numbers and regulators: Direct $q$ -series generating functions for class numbers of quadratic fields via mock Maass theta forms (Rhoades et al., 2011, Li et al., 2023).
Partitions and ranks: Asymptotics and exact formulas for coefficients of partition and rank statistics leverage Maass–Poincaré series and spectral averages (Ahlgren et al., 2018, Li et al., 2013).
Quantum modularity: Renormalization interpretations and quantum modular forms provide frameworks for the partial theta–mock theta correspondence and equidistribution phenomena (Li et al., 2013, Andersen et al., 2016).
Automorphic representations: Shadow correspondences and theta lifts bridge harmonic Maass forms and automorphic representations of higher rank groups (Langlands functoriality) (Pierre, 2012, Alfes-Neumann et al., 2017).

Open problems include the full classification of “ghost terms” in renormalized series, Hecke-type expansions for new families, and deeper understanding of the geometry underlying modular and quantum modular phenomena (Li et al., 2013).

In summary, mock Maass theta functions encode an overview of analytic, modular, algebraic, and arithmetic structures. They provide a comprehensive framework for generalizing, classifying, and explicitly constructing modular objects whose Fourier coefficients and transformation laws reveal profound connections to classical questions in number theory, automorphic forms, and $q$ -hypergeometric series.