A Rademacher exact type formula for pod$_2(n)$

Abstract: In this paper, we calculate an exact formula for the number of partitions of a natural number $n$, where the largest part is even and no odd parts appears more than two times. The generating functions of the number of these partitions is a mixed mock modular form of weight 0. In order to obtain the formula we apply an extended version of the circle method, during which we need to bound Kloosterman sums and similar exponential sums as well as Mordell-type integrals.

• The paper derives a Rademacher-type exact formula for pod₂(n) using an extended circle method and modular transformation techniques.
• It establishes analytic bounds on generalized Kloosterman sums and expresses the generating function in terms of eta-quotients and Mordell integrals.
• The approach allows precise computation and asymptotic analysis of pod₂(n), extending classical partition theory to mixed mock modular contexts.

Rademacher-Type Exact Formula for $\mathrm{pod}_2(n)$

Introduction

The paper "A Rademacher exact type formula for $\mathrm{pod}_2(n)$" (2604.18241) addresses the enumeration of partitions of integers with constrained multiplicities on odd parts and parity conditions on the largest part. Specifically, $\mathrm{pod}_2(n)$ counts partitions of $n$ where:

• No odd part occurs more than twice,
• The largest part is even.

The generating function for $\mathrm{pod}_2(n)$ is shown to be a mixed mock modular form of weight zero, linked intricately to Ramanujan's third-order mock theta function $\rho(q)$. The primary achievement is an analytic, Rademacher-type exact formula for $\mathrm{pod}_2(n)$, leveraging an extended circle method and deep analytic bounds on Kloosterman-type sums and Mordell integrals.

Framework and Background

Partition Constraints and Generating Functions

The domain is a subset of integer partitions, further specified by imposing a frequency constraint on odd integers (at most twice) and an even requirement on the maximal part. The function $\mathrm{POD}_2(q)$ is constructed as:

$$\mathrm{POD}_2(q) = \sum_{n = 0}^\infty \mathrm{pod}_2(n) q^n$$

which is connected to Ramanujan's third-order mock theta function $\rho(q)$ via explicit eta-quotient factorizations. The analysis relates $\mathrm{pod}_2(n)$0 to both the classical partition generating functions and mock modular objects, facilitating the import of modular transformation theory.

Mock Modular Forms and Modular Transformations

The generating function for the relevant partitions falls into the class of mixed mock modular forms with weight zero, incorporating both genuine modular and mock components. The paper methodically rewrites $\mathrm{pod}_2(n)$1 using Ramanujan's identities to express it as a linear combination of functions with well-understood modular transformations, notably $\mathrm{pod}_2(n)$2 (another mock theta function) and eta-quotients. These modular properties are essential for successful deployment of the circle method and for controlling the transformation behavior at cusps.

Circle Method Construction

The Rademacher circle method is adapted to the non-classical mock modular context by decomposing the generating function and analyzing its Fourier coefficients using Cauchy's integral formula along carefully chosen contours. The procedure is tailored for each case according to the residue class of the denominator modulus with respect to $\mathrm{pod}_2(n)$3, employing Farey sequences and exploiting congruence relations for effective partitioning of the integral.

Four distinct summation cases emerge, corresponding to gcd$\mathrm{pod}_2(n)$4. In each, modular transformation formulas and mock-theta expansions lead to integral representations involving eta-quotients, Mordell-type integrals, and Kloosterman sums.

Kloosterman Sums, Bounds, and Integral Evaluations

A core technical aspect is the analysis of intricate exponential sums---generalized Kloosterman sums---arising from modular transformation multipliers and congruence constraints. The paper systematically classifies and evaluates these, including incomplete and shifted variants, and establishes their bounds:

$\mathrm{pod}_2(n)$5

for suitable families indexed by modulus and congruence conditions. These bounds are critical for ensuring convergence of the summations and for controlling contributions from the non-principal arcs in the circle method decomposition.

Mordell-type integrals with Bessel functions appear in the principal parts, requiring further analytic control. Exact integral reductions are provided in terms of modified Bessel functions and explicit integrals over $\mathrm{pod}_2(n)$6, yielding the form:

$\mathrm{pod}_2(n)$7

Main Exact Formula

The principal output is a multi-term Rademacher-type series for $\mathrm{pod}_2(n)$8:

$\mathrm{pod}_2(n)$9

where each term is an explicit sum over moduli $\mathrm{pod}_2(n)$0, with intricate Kloosterman coefficients and Mordell integrals encoded via $\mathrm{pod}_2(n)$1 and $\mathrm{pod}_2(n)$2, all subject to divisor conditions and bounds proved earlier. Notably, only four terms (for the respective residue classes) survive as $\mathrm{pod}_2(n)$3; several principal parts cancel due to analytic identities for the Kloosterman sums and their signs.

Numerical and Analytical Validity

The derived formula admits both theoretical and numerical verification. Analytic bounds ensure absolute convergence for each term in the infinite sum. Asymptotic behavior analysis reveals the dominant exponential rate for large $\mathrm{pod}_2(n)$4, and the exact identity with observed combinatorial counts (as given for small $\mathrm{pod}_2(n)$5 in the introduction) supports correctness.

Implications and Outlook

Theoretical Impact

The work generalizes the classical Rademacher series for partitions into the field of mock modular forms of weight zero. By handling mixed mock modular objects related to partition functions with combinatorial constraints, it opens the avenue for exact series representations in broader contexts than previously accessible---notably for partition functions associated with mock theta functions and their mixed variants.

The analytic control over generalized Kloosterman sums, including incomplete and shifted variants, provides a template for tackling similar sums in other mock modular settings.

Practical Extensions

The exact formula enables precise computation of $\mathrm{pod}_2(n)$6 for arbitrarily large $\mathrm{pod}_2(n)$7 and provides asymptotic expressions amenable to further analysis. This can inform computational studies in combinatorics and connect with modular representation theory problems where similar partition constraints arise.

The approach and technical machinery are transferable to other partition enumeration problems where generating functions exhibit mock modular or mixed modular behavior. Future work may address more complex constraints (higher multiplicity bounds, parity conditions, etc.) and explore links between partition problems and quantum modular forms, analytic number theory, and automorphic representation theory.

Conclusion

The paper establishes a Rademacher-type exact formula for $\mathrm{pod}_2(n)$8, the number of partitions with restricted odd parts and even maximal part, synthesizing modular transformation theory, mock modular forms, and analytic number theory. The result is a convergent, explicit series involving Kloosterman sums and Mordell integrals, with rigorous analytic bounds supporting the formula's validity. The framework generalizes classical partition enumeration to problems involving mock modular objects and opens potential for further developments in analytic enumerative combinatorics and modular form theory.