Rademacher Expansion of Modular Integrals

• Rademacher expansion of modular integrals is a method that expresses modular invariants as explicit convergent series over modular images, incorporating arithmetic factors like Kloosterman and Dedekind sums.
• It employs specialized functions—including incomplete gamma, I‑Bessel, and J‑Bessel functions—to regularize and reconstruct Fourier coefficients and modular integral representations even in critical weight regimes.
• The framework extends to mixed mock, vector-valued, and physical partition functions, providing precise control over convergence, polar contributions, and modular completions.

Searching arXiv for relevant papers on Rademacher expansions, modular integrals, and related contexts. Rademacher expansion of modular integrals denotes a family of exact or convergent reorganizations in which modularly invariant or modularly covariant quantities are expressed as sums over modular images, cusps, or Ford-circle data, with arithmetic factors such as Kloosterman sums or Dedekind sums and analytic kernels built from incomplete gamma functions, Bessel functions, or Eichler integrals. In the classical setting, Rademacher sums reconstruct Fourier coefficients of modular forms from polar data; in modern usage, the same architecture is extended to weakly holomorphic, mock, mixed mock, vector-valued, and non-holomorphic modular objects, and also to actual integrals over modular fundamental domains (Cheng et al., 2012, Baccianti et al., 23 Jan 2025).

1. Classical construction and regularized Rademacher sums

The basic setup begins with a Fuchsian subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{R})$ containing $-I$, commensurable with $\mathrm{SL}_2(\mathbb{Z})$, with cusp width $h$ at infinity defined by $\Gamma_\infty=\langle T^h,-I\rangle$, where $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$ (Cheng et al., 2012). For $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$ one sets $j(\gamma,\tau)=(c\tau+d)^{-2}$, and a multiplier system $\psi:\Gamma\to\mathbb{C}^\times$ of weight $\mathrm{SL}_2(\mathbb{R})$0 satisfies

$\mathrm{SL}_2(\mathbb{R})$1

with the consistency condition $\mathrm{SL}_2(\mathbb{R})$2, where $\mathrm{SL}_2(\mathbb{R})$3 (Cheng et al., 2012). The induced $\mathrm{SL}_2(\mathbb{R})$4-action is

$\mathrm{SL}_2(\mathbb{R})$5

Given $\mathrm{SL}_2(\mathbb{R})$6 with $\mathrm{SL}_2(\mathbb{R})$7, the seed is $\mathrm{SL}_2(\mathbb{R})$8. The non-trivial right-cosets $\mathrm{SL}_2(\mathbb{R})$9 are ordered by the truncation

$-I$0

and one sums over $-I$1 before taking $-I$2 (Cheng et al., 2012). For $-I$3 one requires the regularization factor

$-I$4

with $-I$5 when $-I$6, while $-I$7 for $-I$8 or $-I$9 (Cheng et al., 2012). For $\mathrm{SL}_2(\mathbb{Z})$0 this simplifies to

$\mathrm{SL}_2(\mathbb{Z})$1

The regularized Poincaré or Rademacher sum is then

$\mathrm{SL}_2(\mathbb{Z})$2

with a constant-term correction for $\mathrm{SL}_2(\mathbb{Z})$3 and $\mathrm{SL}_2(\mathbb{Z})$4 (Cheng et al., 2012). For $\mathrm{SL}_2(\mathbb{Z})$5 these are ordinary Poincaré series; for $\mathrm{SL}_2(\mathbb{Z})$6, convergence follows from Niebur-type results; and for the delicate range $\mathrm{SL}_2(\mathbb{Z})$7, convergence is controlled through Selberg’s Kloosterman zeta function, its analytic continuation, and bounds of Goldfeld–Sarnak and Pribitkin (Cheng et al., 2012).

The associated coefficient functions are the Rademacher series

$\mathrm{SL}_2(\mathbb{Z})$8

and, when $\mathrm{SL}_2(\mathbb{Z})$9 and $h$0, the analytic kernel is governed by the $h$1-Bessel function (Cheng et al., 2012). In the canonical normalization much used in the literature,

$h$2

valid when $h$3 and $h$4 (Cheng et al., 2012).

2. Fourier expansions, shadows, and modular integrals in the Eichler sense

Assuming convergence, the Rademacher sum has Fourier expansion

$h$5

with $h$6 (Cheng et al., 2012). For $h$7, this gives the Fourier expansion of a weakly holomorphic modular form or of a mock modular form, depending on $h$8 and $h$9.

For $\Gamma_\infty=\langle T^h,-I\rangle$0, the natural completion is

$\Gamma_\infty=\langle T^h,-I\rangle$1

where $\Gamma_\infty=\langle T^h,-I\rangle$2 is the shadow of weight $\Gamma_\infty=\langle T^h,-I\rangle$3 with multiplier $\Gamma_\infty=\langle T^h,-I\rangle$4 (Cheng et al., 2012). Equivalently, modular invariance is restored under the twisted action

$\Gamma_\infty=\langle T^h,-I\rangle$5

The shadow itself is given by a dual Rademacher sum,

$\Gamma_\infty=\langle T^h,-I\rangle$6

and the coefficient grids satisfy Zagier duality,

$\Gamma_\infty=\langle T^h,-I\rangle$7

which exchanges weight $\Gamma_\infty=\langle T^h,-I\rangle$8 with the dual weight $\Gamma_\infty=\langle T^h,-I\rangle$9 and transposes the coefficient array (Cheng et al., 2012).

This duality is closely tied to Eichler integrals. For a cusp form $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$0 of weight $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$1, its Eichler integral is

$T=\begin{bmatrix}1&1\0&1\end{bmatrix}$2

The paper formulates Eichler duality term-by-term via the dual Rademacher series, so that the completion or Eichler integral is computed directly from dual coefficients (Cheng et al., 2012). In this sense, one established meaning of “modular integral” in the Rademacher literature is an Eichler integral or automorphic integral rather than an integral over the modular fundamental domain.

This distinction becomes explicit in the vector-valued theory. In the rational-weight setting, an automorphic integral of weight $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$3 with multiplier $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$4 is a holomorphic vector-valued function $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$5 such that there exists a cusp form $T=\begin{bmatrix}1&1\0&1\end{bmatrix}$6 with

$T=\begin{bmatrix}1&1\0&1\end{bmatrix}$7

where

$T=\begin{bmatrix}1&1\0&1\end{bmatrix}$8

(Whalen, 2014). In that framework, vector-valued Rademacher sums provide bases for spaces of automorphic integrals, and the non-holomorphic completion is again obtained by subtracting the Eichler integral of the shadow (Whalen, 2014).

A central example of this generalized setting is the mixed mock modular form arising from rank-$T=\begin{bmatrix}1&1\0&1\end{bmatrix}$9 sheaves on $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$0. There the generating functions are expressed in terms of class-number series and $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$1, and the exact coefficient formula is described as the first exact formula for the Fourier coefficients of mixed mock modular forms, with classical Kloosterman–Bessel terms supplemented by explicit corrections coming from the Eichler integral of the shadow (Bringmann et al., 2010). This suggests that, in the mock and mixed mock setting, “Rademacher expansion of modular integrals” naturally includes coefficient formulas in which the modular integral is the Eichler integral required for modular completion.

3. Modular integrals over fundamental domains and the two-dimensional expansion

A second, more literal usage concerns actual modular integrals of non-holomorphic modular functions over the fundamental domain. The 2025 paper formulates this problem for a modular invariant differential $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$2-form

$$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$3

with $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$4 obtained by complexifying the anti-holomorphic variable through $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$5 (Baccianti et al., 23 Jan 2025). The principal value modular integral is

$$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$6

and an $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$7 prescription regulates cusp divergences (Baccianti et al., 23 Jan 2025).

The first step is a holomorphic splitting analogous to the Lorentzian inversion formula in conformal field theory:

$$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$8

with $$\gamma=\begin{bmatrix}a&b\ c&d\end{bmatrix}$$9 (Baccianti et al., 23 Jan 2025). Equivalently,

$j(\gamma,\tau)=(c\tau+d)^{-2}$0

with the contours turned appropriately near $j(\gamma,\tau)=(c\tau+d)^{-2}$1 (Baccianti et al., 23 Jan 2025).

The second step is the genuinely two-dimensional Rademacherization. The split contour is deformed into a sum over Ford circles $j(\gamma,\tau)=(c\tau+d)^{-2}$2 attached to rational points $j(\gamma,\tau)=(c\tau+d)^{-2}$3, with $j(\gamma,\tau)=(c\tau+d)^{-2}$4 integrated along a horizontal contour at large imaginary part (Baccianti et al., 23 Jan 2025). The final exact formula is

$j(\gamma,\tau)=(c\tau+d)^{-2}$5

where $j(\gamma,\tau)=(c\tau+d)^{-2}$6 is the Dedekind sum (Baccianti et al., 23 Jan 2025). This formula expresses the original modular integral in terms of local data near the Lorentzian cusps $j(\gamma,\tau)=(c\tau+d)^{-2}$7 and $j(\gamma,\tau)=(c\tau+d)^{-2}$8.

For integrands of the form

$j(\gamma,\tau)=(c\tau+d)^{-2}$9

only polar terms contribute to the Ford-circle integrals (Baccianti et al., 23 Jan 2025). The resulting universal kernels are oscillatory integrals

$\psi:\Gamma\to\mathbb{C}^\times$0

and

$\psi:\Gamma\to\mathbb{C}^\times$1

$\psi:\Gamma\to\mathbb{C}^\times$2

(Baccianti et al., 23 Jan 2025). The appearance of $\psi:\Gamma\to\mathbb{C}^\times$3 rather than $\psi:\Gamma\to\mathbb{C}^\times$4 reflects the Lorentzian contour geometry of the two-dimensional construction.

The paper emphasizes that this is the exact analogue of the classical one-dimensional Rademacher expansion, but for actual modular integrals of non-holomorphic modular functions. A plausible implication is that the term “Rademacher expansion of modular integrals” now covers two technically distinct but structurally parallel programs: coefficient reconstruction by regularized Poincaré sums, and contour-level evaluation of modular-domain integrals by Ford-circle decompositions.

4. Arithmetic kernels, contour geometry, and convergence mechanisms

Across the literature, the expansion is controlled by a common arithmetic-analytic architecture. On the arithmetic side, single-$\psi:\Gamma\to\mathbb{C}^\times$5 expansions package modular images into Kloosterman-type sums. In the holomorphic or weakly holomorphic case,

$\psi:\Gamma\to\mathbb{C}^\times$6

and the coefficients are assembled from $\psi:\Gamma\to\mathbb{C}^\times$7 and $\psi:\Gamma\to\mathbb{C}^\times$8-Bessel kernels (Cheng et al., 2012). In the two-dimensional modular-integral problem, modular images are weighted instead by phases involving modular inverses $\psi:\Gamma\to\mathbb{C}^\times$9 and Dedekind sums, and the multiplicity function is

$\mathrm{SL}_2(\mathbb{R})$00

with $\mathrm{SL}_2(\mathbb{R})$01 the Rademacher function (Baccianti et al., 23 Jan 2025).

On the analytic side, regularization is indispensable. For one-dimensional Rademacher sums, the lower incomplete gamma factor $\mathrm{SL}_2(\mathbb{R})$02 subtracts the non-convergent part of the seed when $\mathrm{SL}_2(\mathbb{R})$03 (Cheng et al., 2012). In the two-dimensional theory, a principal-value prescription and contour deformation around the cusp play the corresponding role (Baccianti et al., 23 Jan 2025). The same pattern reappears in mixed mock examples, where Mordell-type integrals or Eichler-type integrals supply the non-holomorphic correction required by modularity (Rausch, 20 Apr 2026, Bringmann et al., 2010).

Convergence is established by different mechanisms in different regimes. For $\mathrm{SL}_2(\mathbb{R})$04, Poincaré series converge absolutely; for $\mathrm{SL}_2(\mathbb{R})$05, Niebur-type results give convergence; and for $\mathrm{SL}_2(\mathbb{R})$06, Kloosterman zeta functions and Lipschitz summation yield the delicate estimates, including $\mathrm{SL}_2(\mathbb{R})$07 truncation control in the $\mathrm{SL}_2(\mathbb{R})$08 ordering (Cheng et al., 2012). In the two-dimensional modular-integral setting, a sufficient criterion is the uniform bound

$\mathrm{SL}_2(\mathbb{R})$09

along the Rademacher contours, together with the average estimate

$\mathrm{SL}_2(\mathbb{R})$10

for large $\mathrm{SL}_2(\mathbb{R})$11 (Baccianti et al., 23 Jan 2025).

A useful contrast is provided by negative-weight exact formulas such as the $\mathrm{SL}_2(\mathbb{R})$12 expansion. For $\mathrm{SL}_2(\mathbb{R})$13, the paper specializes the Rademacher-type exact formula to

$\mathrm{SL}_2(\mathbb{R})$14

with the tail bounded by

$\mathrm{SL}_2(\mathbb{R})$15

(Mukherjee, 2024). This is not a modular-domain integral, but it exemplifies the same Rademacher–Bessel principle: exact reconstruction from a polar part, explicit arithmetic sums, and rigorous tail control.

5. Generalizations: mixed mock, Siegel, vector-valued, and physical settings

The modern theory extends well beyond scalar holomorphic forms on $\mathrm{SL}_2(\mathbb{R})$16. In mixed mock modular situations, the exact formulas acquire Eichler or Mordell-type integral corrections. For the partition function $\mathrm{SL}_2(\mathbb{R})$17, the generating series is written as

$\mathrm{SL}_2(\mathbb{R})$18

exhibiting it as a mixed mock modular form of weight $\mathrm{SL}_2(\mathbb{R})$19 (Rausch, 20 Apr 2026). The exact formula then involves Kloosterman-type sums together with the Mordell-type integrals

$\mathrm{SL}_2(\mathbb{R})$20

and

$\mathrm{SL}_2(\mathbb{R})$21

(Rausch, 20 Apr 2026). Here the modular-integral contribution is no longer peripheral: it is part of the exact coefficient formula.

The higher-rank analogue is developed for the reciprocal of the Igusa cusp form $\mathrm{SL}_2(\mathbb{R})$22 on $\mathrm{SL}_2(\mathbb{R})$23. There the Fourier coefficients of $\mathrm{SL}_2(\mathbb{R})$24 are expressed as a regularized sum over residues of quadratic poles, organized by two commuting $\mathrm{SL}_2(\mathbb{R})$25 subgroups inside $\mathrm{SL}_2(\mathbb{R})$26 (Cardoso et al., 2021). The resulting “fine-grained Rademacher-type expansion” contains generalized Kloosterman sums twisted by Jacobi multipliers, as well as three analytic sectors: an $\mathrm{SL}_2(\mathbb{R})$27 polar term, an $\mathrm{SL}_2(\mathbb{R})$28 shadow term, and an $\mathrm{SL}_2(\mathbb{R})$29 Eichler-integral term (Cardoso et al., 2021). The paper explicitly describes this construction as a concrete realization of Rademacher’s philosophy for modular integrals in several complex variables.

In non-holomorphic real-analytic settings, the 2d CFT partition function furnishes another generalization. The spectral density is written as

$\mathrm{SL}_2(\mathbb{R})$30

with Kloosterman sums $\mathrm{SL}_2(\mathbb{R})$31 and a closed $\mathrm{SL}_2(\mathbb{R})$32 density obtained by contour deformation and inverse Laplace transforms (Alday et al., 2019). The expansion converges for spin $\mathrm{SL}_2(\mathbb{R})$33, while $\mathrm{SL}_2(\mathbb{R})$34 is obstructed by the pole of the Eisenstein series at $\mathrm{SL}_2(\mathbb{R})$35 (Alday et al., 2019). Although this is not phrased as a modular-domain integral over $\mathrm{SL}_2(\mathbb{R})$36, it again exemplifies the extension of Rademacher methods to real-analytic modularly invariant objects.

Vector-valued rational-weight Rademacher sums provide yet another axis of generalization. For normal multipliers $\mathrm{SL}_2(\mathbb{R})$37 of finite image, the coefficient formulas involve matrix-valued Kloosterman sums and $\mathrm{SL}_2(\mathbb{R})$38-Bessel functions, and the resulting Rademacher sums span spaces of vector-valued automorphic integrals with prescribed principal parts (Whalen, 2014). This suggests that the Rademacher expansion of modular integrals is naturally compatible with multiplier systems, nontrivial cusp widths, and vector-valued shadows.

6. Canonical examples and applications

The basic scalar model is the $\mathrm{SL}_2(\mathbb{R})$39-function. For $\mathrm{SL}_2(\mathbb{R})$40, $\mathrm{SL}_2(\mathbb{R})$41, $\mathrm{SL}_2(\mathbb{R})$42, and $\mathrm{SL}_2(\mathbb{R})$43, one has

$\mathrm{SL}_2(\mathbb{R})$44

so $\mathrm{SL}_2(\mathbb{R})$45 (Cheng et al., 2012). The coefficients satisfy

$\mathrm{SL}_2(\mathbb{R})$46

with the classical Kloosterman sum $\mathrm{SL}_2(\mathbb{R})$47 (Cheng et al., 2012). A simple modular integral built from $\mathrm{SL}_2(\mathbb{R})$48 is

$\mathrm{SL}_2(\mathbb{R})$49

and unfolding gives

$\mathrm{SL}_2(\mathbb{R})$50

which can then be rewritten using the Rademacher expansion of $\mathrm{SL}_2(\mathbb{R})$51 (Cheng et al., 2012). This is the rigorous form of the “Farey tail” evaluation used in the physics literature.

The 2025 two-dimensional theory yields the first analytic formulas for several one-loop string quantities. For the bosonic string partition function,

$\mathrm{SL}_2(\mathbb{R})$52

the real part is expressed as a convergent Dedekind-sum/Bessel series,

$\mathrm{SL}_2(\mathbb{R})$53

with numerical value

$\mathrm{SL}_2(\mathbb{R})$54

and exact imaginary part

$\mathrm{SL}_2(\mathbb{R})$55

(Baccianti et al., 23 Jan 2025). For the $\mathrm{SL}_2(\mathbb{R})$56 heterotic string, the same method gives

$\mathrm{SL}_2(\mathbb{R})$57

leading to a positive cosmological constant $\mathrm{SL}_2(\mathbb{R})$58 (Baccianti et al., 23 Jan 2025).

Applications also extend to moonshine. In monstrous moonshine, each McKay–Thompson series is recovered by a weight-$\mathrm{SL}_2(\mathbb{R})$59 Rademacher sum,

$\mathrm{SL}_2(\mathbb{R})$60

and this equivalence is described as precisely the genus-zero property (Cheng et al., 2012). In Mathieu moonshine,

$\mathrm{SL}_2(\mathbb{R})$61

with shadow proportional to $\mathrm{SL}_2(\mathbb{R})$62 and completion given by the Eichler integral formula (Cheng et al., 2012). In umbral moonshine, vector-valued Rademacher sums produce the corresponding mock modular forms with specified shadows (Cheng et al., 2012).

A further physically motivated application appears in the study of sunrise integrals, where eta quotients and Eichler integrals govern multi-loop amplitudes. At four and six loops, the on-shell integrals are periods of modular forms of weights $\mathrm{SL}_2(\mathbb{R})$63 and $\mathrm{SL}_2(\mathbb{R})$64 given by Eichler integrals of eta quotients, while weakly holomorphic eta quotients determine quasi-periods (Acres et al., 2018). The paper gives a Rademacher sum formula for Fourier coefficients of an eta quotient that is a Hauptmodul for $\mathrm{SL}_2(\mathbb{R})$65 and generalizes it to many other levels (Acres et al., 2018).

7. Conceptual scope, terminology, and open directions

The literature supports two technically distinct meanings of modular integral. One is the classical Eichler or automorphic integral attached to a cusp form shadow, as in mock modular completions and vector-valued automorphic integrals (Cheng et al., 2012, Whalen, 2014, Bringmann et al., 2010). The other is an actual integral over a modular fundamental domain, as in

$\mathrm{SL}_2(\mathbb{R})$66

or its non-holomorphic two-variable generalization (Cheng et al., 2012, Baccianti et al., 23 Jan 2025). The shared term “Rademacher expansion of modular integrals” is therefore somewhat overloaded. The sources do not present this as a contradiction; rather, they show that both notions are organized by the same structural data: polar information, modular images, arithmetic phases, and special-function kernels.

Several common misconceptions are corrected by the modern theory. First, Rademacher expansions are not restricted to scalar holomorphic modular functions of weight $\mathrm{SL}_2(\mathbb{R})$67; the cited works cover arbitrary real or rational weights, multiplier systems, vector-valued forms, mixed mock modular forms, meromorphic Jacobi and Siegel objects, and real-analytic partition functions (Cheng et al., 2012, Whalen, 2014, Cardoso et al., 2021, Alday et al., 2019). Second, the special-function kernel is not universally $\mathrm{SL}_2(\mathbb{R})$68; the two-dimensional Lorentzian modular-integral expansion produces $\mathrm{SL}_2(\mathbb{R})$69 kernels instead (Baccianti et al., 23 Jan 2025). Third, convergence is not automatic in the critical weight range or at spin $\mathrm{SL}_2(\mathbb{R})$70; regularization by incomplete gamma functions, Eisenstein subtraction, principal-value prescriptions, or theta-constant subtraction can be essential (Cheng et al., 2012, Alday et al., 2019, Cardoso et al., 2021).

Open directions are highlighted explicitly in several works. The 2012 survey lists open problems in moonshine and in the physics interpretation of Rademacher sums (Cheng et al., 2012). The Siegel-modular construction notes that a first-principles “gravity path integral” derivation of the required regulators remains open (Cardoso et al., 2021). The 2025 two-dimensional theory suggests extension to vector-valued modular forms, Maass-type non-holomorphic functions, modular graph functions, and higher genus, where suitable generalizations of Ford circles and Farey sequences would be required (Baccianti et al., 23 Jan 2025). This suggests that the subject is best understood not as a single formula, but as a broad exact framework for reorganizing modular data—coefficients, shadows, and actual modular-domain integrals—into arithmetic series controlled by polar behavior at cusps.