Rademacher-Type Formulas

• Rademacher-type formulas are exact series representations for Fourier coefficients of modular and mock modular forms, fundamentally used to compute partition functions.
• They employ the circle method, modular transformations, and arithmetic sums (such as Dedekind and Kloosterman sums) combined with Bessel integrals.
• These formulas extend to diverse partition constraints and even to geometric and PDE contexts, enabling precise numerical evaluations and asymptotic analyses.

Rademacher-type formulas are exact, absolutely convergent series representations for Fourier coefficients of modular and mock modular forms, most classically for integer partition functions and their generalizations. They arise from deep interactions between analytic number theory and the theory of modular forms, particularly through the circle method, and have been systematically extended to a broad range of partition statistics, generalized combinatorial sequences, and even geometric and PDE contexts.

1. Classical Rademacher Formula: The Partition Function

Rademacher's formula gives a convergent expression for the partition function $p(n)$, enumerating the number of ways $n$ can be expressed as a sum of positive integers. Starting from the generating function $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$ and using a contour integral, the function is analyzed near its singularities at the roots of unity. Through Farey/Ford circle decomposition and modular transformation of the Dedekind $\eta$-function, each local contribution is written as a sum involving Kloosterman-type sums and Laplace-type (Bessel) integrals. The final formula is:

$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$

where $$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$ involves the Dedekind sum $s(h,k)$, and $I_{3/2}$ is a modified Bessel function of order $3/2$ (Kong et al., 2023). The series converges absolutely with error term $O(N^{-1/2})$ for level-$n$0 truncation. The dominant asymptotic term recovers the Hardy–Ramanujan exponential growth law.

2. Generalizations: Colored, Regular, and Restricted Partition Functions

The Rademacher-type method extends to families such as $n$1-colored partitions, $n$2-regular partitions, partitions without certain multiplicities, and combinations thereof. For $n$3-colored $n$4-regular partitions, with generating function

$n$5

the exact Rademacher-type formula for $n$6 includes generalized Kloosterman–Dedekind sums, Bessel functions, and explicit combinatorial coefficients:

$n$7

where $n$8 and detailed arithmetic combinatorics defines $n$9 (Agarwal et al., 8 Nov 2025). Specializations recover the classical Rademacher formula, Hagis’s formula for $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$0-regular partitions, and formulas for partitions into distinct parts.

3. Rademacher-Type Formulas for Partitions with Multiplicative or Combinatorial Constraints

Numerous classical and modern results employ the Rademacher circle method for partitions subject to multiplicity, congruence, or combinatorial constraints:

• $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$1-regular partitions (parts not divisible by relatively prime $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$2): the main theorem in (Laughlin et al., 2019) gives

$F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$3

with explicitly defined arithmetic coefficients involving Dedekind sums and eta-multiplier products.

• Partitions without sequences, or with other pattern restrictions: For instance, the function $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$4 (partitions with no two consecutive parts) admits an exact formula as a sum over three cusp types, each term involving Kloosterman sums and real Mordell- or Bessel-type integrals, reflecting the mixed modular and mock-modular nature of the generating function (Bridges et al., 2023).
• Partitions with parity/multiplicity conditions: For $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$5 (partitions with even largest part, each odd part at most twice), the Rademacher-type expansion decomposes into four infinite sums corresponding to $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$6, each with distinct Kloosterman–Mordell–Bessel integrals (Rausch, 20 Apr 2026).
• Overpartition and cubic overpartition formulas demonstrate similar structure, including combinations of Bessel functions, multipliers, and sum-over-coset decompositions, together with explicit error bounds and eventual applications to log-concavity and Turán inequalities (Sills, 2013, Agarwal et al., 27 Sep 2025).

4. Methodological Foundation: The Circle Method and Modular Transformations

All Rademacher-type formulas fundamentally rely on the analytic circle method. The typical process is:

• Express the coefficient as a contour integral of the generating function.
• Decompose the contour along Farey arcs or Ford circles near rationals $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$7 in the complex unit disc or modular domain.
• Apply modular (often Dedekind–eta or mock theta) transformation laws to capture local singularity structure at each cusp.
• Expand local contributions as series involving arithmetic sums (Kloosterman sums with multipliers), with the main exponential and oscillatory part expressed as Bessel or, in the mock-modular/non-holomorphic case, Mordell-type integrals.
• Sum over cusps and, where necessary, sum over additional combinatorial parameters (e.g., for higher-order regular or colored partitions).
• Prove absolute convergence and provide effective error bounds for truncations, typically invoking Weil-Estermann or sharper bounds for Kloosterman sums.

This pipeline admits extensive generalization as long as precise modular and analytical properties of the generating function (modular, mock modular, mixed weight) are available (Kong et al., 2023, Agarwal et al., 8 Nov 2025, Rausch, 20 Apr 2026).

5. Applications: Partition Asymptotics, Inequalities, and Beyond

Rademacher-type formulas provide both exact evaluation and asymptotic growth for partition-theoretic and related combinatorial functions:

• Leading-term asymptotics recover classical exponential laws (e.g., $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$8) and supply main exponents and constants for generalized statistics (Kong et al., 2023, Sills, 2013, Agarwal et al., 8 Nov 2025).
• Precise remainder bounds in these series enable proofs of log-concavity, Turán inequalities, and (generalized) Newton and Jensen hyperbolicity criteria for partition functions and their polynomial analogues (Agarwal et al., 8 Nov 2025, Agarwal et al., 27 Sep 2025).
• Exact formulas allow high-precision numerical evaluation: five $F(z)=\prod_{m=1}^\infty(1-z^m)^{-1}$9-terms are often sufficient for 12-digit agreement with medium-sized $\eta$0 values (Agarwal et al., 8 Nov 2025).
• The analytic machinery extends to sheaf-counting in algebraic geometry and to diverse combinatorial congruence classes, overpartitions, or pattern-avoidance statistics.

6. Extensions and Non-Partition Contexts: Rademacher-Type Theorems in Analysis

Beyond combinatorics, "Rademacher-type" also refers to differentiability theorems analogous to Rademacher’s theorem on Lipschitz differentiability, extended to geometric measure-theoretic and PDE settings.

In the context of the transport equation for currents (integral and normal $\eta$1-currents in $\eta$2), Rademacher-type differentiability theorems provide conditions under which an absolutely continuous path $\eta$3 admits an a.e.-defined geometric derivative (a driving vector field $\eta$4), generalizing classical differentiability for scalar-valued functions:

• The strong Rademacher-type theorem: If the path of boundaryless $\eta$5-currents is absolutely continuous in the flat norm and a negligible-criticality (NC) condition holds, then there exists $\eta$6 so that the PDE $\eta$7 holds almost everywhere in the distributional sense (Bonicatto et al., 2022).
• The critical set and Sard property relate differentiability to the geometric singularities and rectifiability structure of the evolving currents.
• Explicit counterexamples (the "Flat Mountain") demonstrate sharpness of hypotheses in this non-scalar, geometric context.

This analytic generalization links Rademacher-type theorems to the theory of regular Lagrangian flows, geometric measure theory, and transport equations in mathematical physics.

7. Summary Table: Rademacher-Type Series for Partition Families

Partition Type	Main Arithmetic Sum	Bessel Function
Ordinary partitions $\eta$8	$\eta$9 (Dedekind sum)	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$0
$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$1-colored, $$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$2-regular	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$3	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$4
$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$5-regular partitions	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$6 (eta ratios)	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$7
Cubic overpartitions	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$8	$$p(n) = \sum_{k=1}^\infty R_k(n),\quad R_k(n) = \frac{1}{\pi\sqrt{2}}\,A_k(n)^{-1}\,k^{-1/2}\,I_{3/2}\left(\frac{\pi}{k}\sqrt{\frac{2}{3}(n-1/24)}\right),$$9
No-odd-repeat partitions, overpart.	$$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$0	$$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$1
Partitions without sequences	$$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$2 (cusps)	Mordell/Bessel integrals
pod$$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$3, "mixed" cases	$$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$4, etc.	Integrated $$A_k(n)=\sum_{(h,k)=1} \exp(\pi i\,s(h,k)-2\pi i n h/k)$$5, etc.

The precise form of each series depends on combinatorial constraints, properties of the generating function, and the modular or mock modular transformations involved.

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Rademacher-type formulas thus represent a unifying analytic framework for the exact calculation and asymptotic analysis of partition-theoretic objects, providing powerful explicit series and revealing deep connections among modular forms, arithmetic combinatorics, and analysis (Kong et al., 2023, Agarwal et al., 8 Nov 2025, Laughlin et al., 2019, Rausch, 20 Apr 2026, Bonicatto et al., 2022).