Harmonic Maass Forms Summation Formula

• Harmonic Maass forms are nonholomorphic modular objects whose Fourier coefficients are connected via kernel-weighted summation identities.
• The summation formulas employ analytic tools such as L-series, Mellin transforms, and modified Bessel functions to relate dual coefficients and arithmetic invariants.
• Applications of these formulas include new identities for the partition function, Riesz means for cusp forms, and links to mock modular forms and singular moduli.

A summation formula for harmonic Maass forms is a transformation or explicit identity connecting a sum of the Fourier coefficients of a harmonic Maass form—often weighted by an analytic kernel such as a special function—with another sum, typically involving dual coefficients arising from an involutive automorphism (such as the Fricke operator), or arithmetic quantities like traces of singular moduli. Such formulas provide powerful analytic tools and deep arithmetic insight into modular objects of both integral and half-integral weight, and their paper involves techniques from the theory of $L$-series, Mellin transforms, theta lifts, and regularization procedures.

1. Fourier Expansions and Harmonic Maass Forms

A harmonic Maass form $f$ of weight $k$ with respect to a congruence subgroup $\Gamma_0(N)$ (possibly with Nebentypus character) admits a Fourier expansion

$$f(z) = \sum_{n \geq -n_0} c^+(n) e^{2\pi i n z} + \sum_{n>0} c^-(n) \Gamma(1-k, 4\pi n y) e^{-2\pi i n \bar{z}},$$

where $c^+(n)$ and $c^-(n)$ are the "holomorphic" and "non-holomorphic" Fourier coefficients, and $\Gamma(s,x)$ is the incomplete Gamma function. The analytic and arithmetic paper of these coefficients is central to the understanding of harmonic Maass forms and their summation formulas (Diamantis et al., 26 Sep 2025, Diamantis et al., 2022).

2. L-series Machinery and Functional Equations

Recent advances leverage the theory of $L$-series attached to harmonic Maass forms, defined by integrating the form against a family of suitable test functions $\varphi$: $$L_f(\varphi) = \int_{0}^{\infty} f(iy)\,\varphi(y)\,dy.$$ For the Fourier expansion above, this $L$-series can be written as

$$L_f(\varphi) = \sum_{n \geq -n_0} c^{+}(n) (\mathcal{L}\varphi)(2\pi n) + \sum_{n>0} c^-(n)\int_{0}^{\infty}\varphi(t)(\ldots)dt,$$

where $$(\mathcal{L}\varphi)(s) = \int_{0}^{\infty} e^{-st} \varphi(t)\,dt$$ is the Laplace transform (Diamantis et al., 2021). Such $L$-series possess functional equations mirroring modular transformation properties. For example, for a suitable $\varphi$ and Fricke involution $w_N$ one has

$$L_f(\varphi) = i^k N^{1-k/2} L_{f|_k w_N}(\varphi|_{2-k} w_N),$$

which forms the backbone of the kernel transformation in summation formulas.

3. The Summation Formula: Kernel-Weighted Sums and Bessel Transforms

A central result (Diamantis et al., 26 Sep 2025) is a transformation formula relating sums of Fourier coefficients of a harmonic Maass form weighted by special functions (typically modified Bessel functions) to sums over the dual coefficients associated to the Fricke involution. For a given $C>0$ and $X$, and with $D_n = C + (2\pi n)/\sqrt{N}$,

$$\sum_{n \geq -n_0} c^+(n) K_0\big(2\sqrt{ D_n (C + \sqrt{N} X ) }\big) \pm (\text{non-holomorphic part}) = i^k C^{k/2} \sum_{n \geq -n_0} d^+(n) ( \sqrt{N} X + D_n )^{-k/2} K_k\big(2\sqrt{C(D_n + \sqrt{N} X)} \big) + (\text{additional terms}),$$

where $K_\lambda$ denotes the modified Bessel function of the second kind, and $d^+(n)$ are the Fourier coefficients of $f|_k w_N$. Such formulas arise from explicit Mellin inversion identities: $$\int_{0}^{\infty} e^{-ax - b/x} \frac{dx}{x} = 2 K_0(2\sqrt{ab}),$$ which naturally interpolate between exponential and oscillatory behaviors of modular sums. The choice of kernel (test function) is dictated by convergence and growth properties.

4. Applications: Partition Function and Classical Modular Forms

a) Partition Function

For the weakly holomorphic modular form $f(z) = 1/\eta(24z)$, whose $q$-expansion yields the partition numbers $p(n)$,

$$1/\eta(24z) = q^{-1/24} \sum_{n=0}^\infty p(n) q^{24 n},$$

the summation formula specializes to provide a new transformation identity for $p(n)$ (Diamantis et al., 26 Sep 2025): $$\sum_{n \geq 0} p(n) K_0 \big(2\sqrt{ (2\pi n + \varepsilon) (\pi/12 + \varepsilon + 24 X ) }\big) = (\text{prefactor}) \cdot \sum_{n \geq 0} p(n) (24 X + 2\pi n + \varepsilon)^{-1/2} e^{-2\sqrt{ (24 X + 2\pi n + \varepsilon ) ( \pi/12 + \varepsilon ) }},$$ with appropriate choices of constants. This formulation provides an alternative, kernel-weighted summation for the partition function, sharpening earlier algebraic and analytic trace formulas (Bruinier et al., 2011, Alfes, 2012).

b) Riesz Sums for Cusp Forms

For classical holomorphic cusp forms,

$f(z) = \sum_{n > 0} c(n) e^{2\pi i n z},$

the analogous machinery yields summation formulas for Riesz means, employing Kummer's confluent hypergeometric function $M(s,k,x)$ and suitable kernel functions $$\psi_s(t) = \mathbf{1}_{(0,1)}(t)(1-t)^{s-1} t^{k-s-1}/\Gamma(s)$$. These formulas allow extraction of asymptotic behaviors and exact relations between weighted sums of Fourier coefficients and special values of $L$-series.

5. Generalizations and Associated Summation Phenomena

The kernel-based summation formula is not only limited to forms of integral weight but extends to half-integral weight, vector-valued harmonic Maass forms, and harmonic lifts. Extensions via theta lifts—using Kudla–Millson kernels or Shintani lift machinery—relate these summation formulas to traces of singular moduli, periods of modular functions, and algebraic objects in ring class fields (Bruinier et al., 2011, Alfes-Neumann et al., 2017, Alfes-Neumann et al., 2022). In many instances, such formulas lead to finite algebraic expressions for Fourier coefficients or CM-value traces, connecting automorphic analysis to deep arithmetic.

Regularized integrals and $L$-series approaches provide further generalizations, enabling the formulation of summation identities for nonholomorphic forms, period integrals, and even for forms with nontrivial growth at cusps. For instance, the test-function formalism discussed in (Diamantis et al., 2022, Diamantis et al., 2021) leads to Voronoi-type, regularized summation formulas which are valid uniformly across a broad class of harmonic Maass forms.

6. Significance and Impact

Summation formulas for harmonic Maass forms unify several analytic, arithmetic, and geometric phenomena:

• They provide explicit dualities between sums over Fourier coefficients and their images under modular involutions, with kernels encoding analytic or arithmetic information.
• The use of $L$-series and functional equations motivates analytic continuation and regularization techniques, extending the range of modular and Maass forms for which these formulas are valid.
• Applications include nontrivial identities for the partition function, mock theta functions, and modular $L$-values, with ramifications for combinatorics, the theory of traces of singular moduli, and special values of $L$-functions.
• The techniques generalize classical summation phenomena such as the Voronoi formula, and via Mellin inversion connect to spectral and representation-theoretic frameworks (Bringmann et al., 2016, Jääsaari et al., 2016).

The ongoing development of these methods continues to open new directions: from the algebraic classification of Fourier coefficients as traces (Alfes, 2012, Alfes-Neumann et al., 2022), to the construction of nonholomorphic lifts and regularized period formulas (Jeon et al., 2012, Alfes-Neumann et al., 2017), and their incorporation into the theory of automorphic $L$-functions and periods.

7. Selected Table: Kernel Functions and Their Use in Summation Formulas

Kernel Type	Formula Example	Application
Modified Bessel $K_\nu$	$K_0(2\sqrt{ab})$, $K_k(2\sqrt{ab})$	Dual sum transformations
Laplace Transform	$(\mathcal{L}\varphi)(s)$	$L$-series and regularized sums
Confluent Hypergeometric $M$	$M(s,k,x)$	Riesz means for cusp forms

Analytic kernels are carefully chosen to ensure convergence, Mellin-inversion accessibility, and compatibility with functional equations linking Fourier coefficients to their duals.

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Summation formulas for harmonic Maass forms, primarily through kernel-weighted duality, Mellin and Laplace integral transforms, and $L$-series functional equations, offer a rigorous and flexible toolkit for relating Fourier expansions and arithmetic invariants in the theory of automorphic forms. Their modern development, as exemplified in (Diamantis et al., 26 Sep 2025), systematically extends classical summation techniques, giving analytic control and explicit identities for a much broader class of modular objects.