A Rademacher exact type formula for pod₂(n)
Pith reviewed 2026-05-10 03:42 UTC · model grok-4.3
The pith
An exact formula exists for pod₂(n), the number of partitions of n with even largest part and odd parts repeating at most twice.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We calculate an exact formula for the number of partitions of a natural number n where the largest part is even and no odd part appears more than twice. The generating function is a mixed mock modular form of weight 0. To obtain the formula we apply an extended version of the circle method, during which we bound Kloosterman sums and similar exponential sums as well as Mordell-type integrals.
What carries the argument
The extended circle method applied to the mixed mock modular generating function, yielding a Rademacher-type exact formula after sufficient bounds on the Kloosterman sums and Mordell integrals.
Load-bearing premise
The bounds on Kloosterman sums and Mordell-type integrals are strong enough for the extended circle method to deliver an exact formula.
What would settle it
A direct count of pod₂(20) that differs from the value given by the closed-form expression derived in the paper.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper derives an exact Rademacher-type formula for the partition function pod₂(n), counting partitions of n where the largest part is even and no odd part appears more than twice. The generating function is treated as a mixed mock modular form of weight 0; an extended circle method is applied, with bounds on Kloosterman sums, related exponential sums, and Mordell-type integrals used to show that remainder terms from the Farey dissection vanish identically, yielding a finite exact sum rather than an asymptotic expansion.
Significance. If the stated bounds hold and produce exact vanishing of error terms, the result extends Rademacher's classical exact formula to a new family of partition functions attached to mixed mock modular forms. This would supply a closed-form expression useful for exact computation and further analytic number theory, while demonstrating that the circle method can be adapted to handle the non-holomorphic components without introducing free parameters or fitted constants.
minor comments (3)
- The abstract and title use slightly inconsistent subscript notation (pod₂(n) vs. pod_2(n)); standardize throughout the manuscript and in all displayed formulas.
- Definition of pod₂(n) is given only verbally; include a short table or explicit values for small n (e.g., n=1 to 10) to make the combinatorial condition immediately verifiable.
- The transformation law for the mixed mock modular form is invoked but not restated; a brief self-contained recall of the relevant weight-0 transformation property (including the non-holomorphic correction term) would improve readability for readers outside the immediate subfield.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for the positive assessment, including the accurate summary of our derivation of the exact Rademacher-type formula for pod₂(n) via the extended circle method applied to its mixed mock modular generating function. We are pleased with the recommendation for minor revision and the recognition of the result's potential significance.
Circularity Check
No significant circularity; direct analytic derivation from generating function
full rationale
The paper applies the extended circle method to the mixed mock modular generating function of weight 0 for pod₂(n) to obtain an exact Rademacher-type formula. It derives the necessary bounds on Kloosterman sums, related exponential sums, and Mordell-type integrals to show that remainder terms from the Farey dissection vanish identically, producing a finite exact sum. No step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the transformation properties and non-holomorphic handling follow from the generating function's known properties without circular reduction. The derivation is self-contained against external analytic benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard bounds on Kloosterman sums and exponential sums suffice for the circle method contour integration
- domain assumption The mixed mock modular form of weight 0 admits a circle-method expansion that converges to an exact formula
Reference graph
Works this paper leans on
-
[1]
George E. Andrews. On the theorems of Watson and Dragonette for Ramanujan’s Mock Theta Functions. American Journal of Mathematics, 88(2):454–490, 1966
work page 1966
-
[2]
Andrews.The Theory of Partitions
George E. Andrews.The Theory of Partitions. Encyclopedia of Mathematics and Its Applications. Addison-Wesley, Reading, Mass. : London, 1976
work page 1976
-
[3]
George E. Andrews and Bruce C. Berndt.Ramanujan’s Lost Notebook Part V. Springer, Cham, Switzer- land, 2018
work page 2018
-
[4]
On restricted partitions, unimodal sequences, and concave compositions.to be published, 2026
Koustav Banerjee and Kathrin Bringmann. On restricted partitions, unimodal sequences, and concave compositions.to be published, 2026
work page 2026
-
[5]
Walter Bridges and Kathrin Bringmann. A rademacher-type exact formula for partitions without se- quences.The Quarterly Journal of Mathematics, 75 (1):197–217, 2024
work page 2024
-
[6]
Kathrin Bringmann and Karl Mahlburg. An extension of the hardy-ramanujan circle method and appli- cations to partitions without sequences.American Journal of Mathematics, 133:1151 – 1178, 2011
work page 2011
-
[7]
Kathrin Bringmann and Jan Manschot. From sheaves on P 2 to a generalization of the rademacher expan- sion.American Journal of Mathematics, 135:1039–1065, 2013
work page 2013
-
[8]
Theodor Estermann. Vereinfachter Beweis eines Satzes von Kloosterman.Abhandlungen aus dem Mathe- matischen Seminar der Universit¨ at Hamburg, 1:82–98, 1929
work page 1929
-
[9]
Exact formula for 1-lower run overpartitions, 2023
Lukas Mauth. Exact formula for 1-lower run overpartitions, 2023
work page 2023
-
[10]
Hans Rademacher. A convergent series for the partition functionp(n).Proceedings of the National Academy of Sciences (PNAS), 23 (2):78–84, 1937
work page 1937
-
[11]
Hans A. Rademacher. The fourier coefficients of the modular invariant j(τ).American Journal of Mathe- matics, 60:501, 1938
work page 1938
-
[12]
Hans A. Rademacher. On the expansion of the partition function in a series.Annals of Mathematics, 44:416–422, 1943
work page 1943
-
[13]
Hans A. Rademacher and Herbert S. Zuckerman. On the fourier coefficients of certain modular forms of positive dimension.Annals of Mathematics, 39:433–462, 1938
work page 1938
-
[14]
Hans Sali´ e. Zur Absch¨ atzung der Fourierkoeffizienten ganzer Modulformen.Mathematische Zeitschrift, 36:263–278, 1933
work page 1933
-
[15]
Temme.Asymptotic Methods for Integrals, volume 6 ofSeries in Analysis
Nico M. Temme.Asymptotic Methods for Integrals, volume 6 ofSeries in Analysis. World Scientific Pub- lishing, 2015. Department of Mathematics and Computer Science, Division of Mathematics, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany Email address:krausch1@uni-koeln.de
work page 2015
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.