Power partitions and Khinchin families

Jos\'e L. Fern\'andez; V\'ictor J. Maci\'a

arxiv: 2602.18575 · v2 · submitted 2026-02-20 · 🧮 math.PR · math.CV· math.NT

Power partitions and Khinchin families

Jos\'e L. Fern\'andez , V\'ictor J. Maci\'a This is my paper

Pith reviewed 2026-05-15 20:32 UTC · model grok-4.3

classification 🧮 math.PR math.CVmath.NT

keywords partitions into powersgenerating functionsKhinchin familiesGaussianityasymptotic formulasHayman's methodBáez-Duarte criterionanalytic number theory

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The pith

The generating function for partitions into k-th powers is strongly Gaussian in the Báez-Duarte sense.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the ordinary generating function for the number of partitions of an integer into sums of kth powers satisfies the strong Gaussianity property inside the Khinchin families framework. If this holds, Hayman's theorem on Gaussian power series supplies the classical asymptotic formula for the partition count p_k(n) at once, with explicit constants derived from the mean and variance. A reader would care because the result recasts a classical analytic-number-theory statement as a direct consequence of a probabilistic criterion rather than a separate saddle-point analysis. The argument verifies the criterion by feeding existing growth bounds into a general Gaussianity test for Khinchin families. This supplies a uniform route to asymptotics for any partition problem whose generating function meets the same bounds.

Core claim

The generating function of partitions into k-th powers is strongly Gaussian in the sense of Báez-Duarte. Within the probabilistic framework of Khinchin families, the Hardy-Ramanujan asymptotic formula for the number p_k(n) of partitions of n into k-th powers reads p_k(n) ~ α_k n^(-(3k+1)/(2k+2)) exp(β_k n^(1/(k+1))), where α_k and β_k are explicit constants depending only on k, then follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity combines a Gaussianity criterion for Khinchin families with bounds of Tenenbaum, Wu and Li on the generating function; the asymptotic formula is recovered by computing asymptotic approximations of

What carries the argument

The Báez-Duarte strong Gaussianity criterion for the Khinchin family attached to the generating function of kth-power partitions, which lets Hayman's formula produce the explicit asymptotic count.

If this is right

The explicit asymptotic p_k(n) ~ α_k n^(-(3k+1)/(2k+2)) exp(β_k n^(1/(k+1))) holds for every fixed positive integer k.
The constants α_k and β_k are obtained directly from the asymptotic mean and variance of the associated Khinchin family.
Hayman's method applies to any partition generating function once the same Gaussianity criterion is verified.
Error terms in the asymptotic follow from the strength of the input bounds on the generating function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same criterion could be checked for generating functions of partitions into values of other polynomials or restricted additive sets.
Strong Gaussianity may yield concentration inequalities or local limit theorems for the distribution of the number of summands.
The framework offers a route to uniform asymptotics across families of partition problems that share comparable growth bounds.

Load-bearing premise

The bounds of Tenenbaum, Wu and Li on the generating function are strong enough to verify the Gaussianity criterion for Khinchin families.

What would settle it

Numerical computation, for a fixed k and sequence of large n, of the third or fourth normalized moment of the probability distribution induced by the coefficients of the generating function, showing failure to approach the corresponding Gaussian moment.

read the original abstract

We prove that the generating function of partitions into $k$-th powers is strongly Gaussian in the sense of B\'aez-Duarte. Within the probabilistic framework of Khinchin families, the Hardy--Ramanujan asymptotic formula for the number~$p_k(n)$ of partitions of~$n$ into $k$-th powers reads \[ p_k(n) \sim \frac{\alpha_k}{n^{(3k+1)/(2k+2)}} \exp\bigl(\beta_k\, n^{1/(k+1)}\bigr), \qquad n \to \infty, \] where $\alpha_k$ and $\beta_k$ are explicit constants depending only on~$k$, then follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity combines a Gaussianity criterion for Khinchin families with bounds of Tenenbaum, Wu and Li on the generating function; the asymptotic formula is recovered by computing asymptotic approximations of the mean and variance of the associated family.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They show the k-power partition generating function is strongly Gaussian in the Báez-Duarte sense and recover the classical asymptotic via Hayman.

read the letter

The paper proves that the generating function for partitions into k-th powers meets the strong Gaussianity condition from Báez-Duarte. This lets them invoke Hayman's theorem directly and obtain the known asymptotic formula for p_k(n) with explicit constants depending only on k. The derivation uses a Gaussianity criterion for Khinchin families together with bounds from Tenenbaum, Wu and Li on the generating function, then extracts the mean and variance asymptotics to finish the argument. Verifying strong Gaussianity for this specific generating function is the new piece, even though the target formula is classical. The logic stays clean because the bounds are external and no fitted parameters appear. The approach gives a probabilistic route to a result that was originally proved by other means. The main soft spot is whether the cited bounds supply the angular uniformity the criterion needs. The Gaussianity check requires control on log P(re^{iθ}) for small θ, not just on the positive real axis, and if the Tenenbaum-Wu-Li estimates lose precision off the real line the verification could have a gap. The abstract indicates they believe the bounds suffice, but the details would need checking. This is for readers already working with Khinchin families or probabilistic methods in partition theory. Someone looking for an alternative derivation of the asymptotic or for examples of the framework applied to concrete generating functions would find it useful. It deserves a serious referee because the central claim is concrete and can be verified against existing literature.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the ordinary generating function P(z) = ∏_{m≥1} (1 - z^{m^k})^{-1} for the number p_k(n) of partitions of n into k-th powers is strongly Gaussian in the Báez-Duarte sense for Khinchin families. The proof verifies a Gaussianity criterion by appealing to bounds of Tenenbaum, Wu and Li on P(z), computes the associated mean and variance asymptotics, and invokes Hayman's theorem to recover the explicit Hardy-Ramanujan-type formula p_k(n) ∼ (α_k / n^{(3k+1)/(2k+2)}) exp(β_k n^{1/(k+1)}).

Significance. If the central claim holds, the work supplies a clean probabilistic derivation of the power-partition asymptotic inside the Khinchin-family framework, showing that existing saddle-point/Tauberian bounds suffice to establish strong Gaussianity without new analytic estimates. This approach is reusable for other partition problems once comparable bounds are available and credits the cited literature explicitly.

major comments (2)

[Proof of strong Gaussianity (central argument combining the criterion with the external bounds)] The verification that the Tenenbaum-Wu-Li bounds imply the precise decay and derivative conditions of the Báez-Duarte Gaussianity criterion (on |log P(r e^{iθ}) - log P(r)| and the second derivative for |θ| ≲ r^{-δ}) is not carried out explicitly. The cited bounds are stated for the positive real radius; without a separate uniformity argument in a sector or disk neighborhood, the invocation of the criterion remains conditional.
[Computation of mean and variance asymptotics] The asymptotic approximations for the mean and variance that feed into Hayman's formula are derived directly from the same bounds. If the angular uniformity required by the Gaussianity criterion fails, these approximations do not automatically extend to the complex neighborhood needed for the final application of Hayman's theorem.

minor comments (2)

[Introduction / statement of the criterion] The precise statement of the Báez-Duarte Gaussianity criterion (including the exact form of the required estimates on log P) should be recalled verbatim or referenced with equation numbers for the reader's convenience.
[Abstract and main theorem statement] Explicit formulas for the constants α_k and β_k in terms of k should be displayed immediately after the asymptotic formula is stated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for identifying the need for greater explicitness in the central argument. We agree that the application of the Báez-Duarte criterion requires a clearer uniformity statement derived from the Tenenbaum-Wu-Li bounds, and we will revise the manuscript to supply the missing details. The two major comments are addressed point by point below.

read point-by-point responses

Referee: The verification that the Tenenbaum-Wu-Li bounds imply the precise decay and derivative conditions of the Báez-Duarte Gaussianity criterion (on |log P(r e^{iθ}) - log P(r)| and the second derivative for |θ| ≲ r^{-δ}) is not carried out explicitly. The cited bounds are stated for the positive real radius; without a separate uniformity argument in a sector or disk neighborhood, the invocation of the criterion remains conditional.

Authors: We acknowledge that the manuscript invokes the Gaussianity criterion after citing the Tenenbaum-Wu-Li bounds but does not spell out the uniformity argument needed to transfer the real-positive estimates to a small angular sector. In the revised version we will insert a dedicated lemma (or subsection) that extracts the required decay |log P(r e^{iθ}) - log P(r)| = O(θ² r^γ) and the second-derivative bound directly from the stated estimates on P(x) for x real and positive, together with a standard contour-shift or Phragmén-Lindelöf argument to control the sector |θ| ≲ r^{-δ}. This will make the verification fully explicit and remove the conditional character of the application. revision: yes
Referee: The asymptotic approximations for the mean and variance that feed into Hayman's formula are derived directly from the same bounds. If the angular uniformity required by the Gaussianity criterion fails, these approximations do not automatically extend to the complex neighborhood needed for the final application of Hayman's theorem.

Authors: The mean and variance asymptotics are indeed obtained from the real-positive bounds on log P(r) and its derivatives. Once the uniformity statement requested in the first comment is established, the same bounds immediately yield the corresponding estimates inside the disk neighborhood required by Hayman’s theorem. In the revision we will add a short paragraph after the uniformity lemma that records this transfer and confirms that the explicit constants α_k and β_k remain valid for the complex application. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent external bounds and standard theorems

full rationale

The paper proves strong Gaussianity of the power-partition generating function by invoking an external Khinchin-family Gaussianity criterion (Báez-Duarte) together with analytic bounds supplied by Tenenbaum-Wu-Li, which are independent of the present authors. The Hardy-Ramanujan-type asymptotic is then recovered directly from Hayman's theorem applied to the resulting mean and variance asymptotics. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing step rests on a self-citation chain. The cited results are externally verifiable and do not incorporate the target Gaussianity statement, so the derivation chain remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on the definition of strong Gaussianity and the Gaussianity criterion for Khinchin families, both taken from prior literature, together with analytic bounds supplied by Tenenbaum, Wu and Li.

axioms (2)

domain assumption Khinchin families admit a Gaussianity criterion that can be checked via bounds on the generating function
Invoked to conclude strong Gaussianity from the cited bounds.
standard math Hayman's asymptotic formula applies to strongly Gaussian power series
Standard result in analytic combinatorics used to extract the coefficient asymptotic.

pith-pipeline@v0.9.0 · 5485 in / 1344 out tokens · 26003 ms · 2026-05-15T20:32:06.864710+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

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B´ aez-Duarte, L.: Hardy–Ramanujan’s Asymptotic Formula for Partitions and the Central Limit Theorem.Adv. Math.125 (1997), no. 1, 114–120

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L., Fern´ andez, P

Cant´ on, A., Fern´ andez, J. L., Fern´ andez, P. and Maci´ a, V. J.: Khinchin families and Hayman class. Comput. Methods Funct. Theory21 (2021), no. 4, 851–904

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L., Fern´ andez, P

Cant´ on, A., Fern´ andez, J. L., Fern´ andez, P. and Maci´ a, V. J.: Khinchin families, set constructions, partitions and exponentials.Mediterr. J. Math.21(2024), article no. 39, 28 pp. 16 J. L. FERN ´ANDEZ AND V. J. MACI ´A

work page 2024
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and Maci´ a V

Cant´ on, A., Fern´ andez, J.L., Fern´ andez, P. and Maci´ a V. J.: Growth of power series with nonnegative coefficients, and moments of power series distributions.Publicacions Matem` atiques70, no 1, (2026), 161–201

work page 2026
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Fern´ andez, J. L. and Maci´ a, V. J.:Large powers asymptotics, Khinchin families and Lagrangian distributions, arXiv:2201.11746 (2022)

work page arXiv 2022
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Hardy, G. H. and Ramanujan, S.: Asymptotic formulae in combinatory analysis.Proc. London Math. Soc.(2)17(1918), 75–115

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Hayman, W. K.: A Generalisation of Stirling’s Formula.J. Reine Angew. Math.196(1956), 67–95

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work page arXiv 2025
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J.:Tesis doctoral/Ph

Maci´ a, V. J.:Tesis doctoral/Ph. D. Thesis: The Theory of Khinchin Families, Universidad Aut´ onoma de Madrid (October 2024). Available at arXiv:2503.14157

work page arXiv 2024
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Maci´ a, V. J.: Some Gaussianity criteria for Khinchin families.J. Math. Anal. Appl.556(2025), 130239

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Murthy, M. V. N., Brack, M., Bhaduri, R. K., and Bartel, J.: Semiclassical analysis of dis- tinct square partitions.Physical Review E98(2018), 052131. DOI: 10.1103/PhysRevE.98.052131. arXiv:1808.05146

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve.98.052131 2018
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C.: Probability and entire functions

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Roth, K. F. and Szekeres, G.: Some asymptotic formulae in the theory of partitions.The Quarterly Journal of Mathematics,5, Issue 1, (1954) pp. 241–259

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Tenenbaum, G., Wu, J., and Li, Y.-L.: Power partitions and saddle-point method.Journal of Number Theory,204(2019), 435–445

work page 2019
[17]

and Li, Y.: Power partitions and saddle-point method.arXiv:1901.02234v4 (2019)

Tenenbaum, G., Wu, J. and Li, Y.: Power partitions and saddle-point method.arXiv:1901.02234v4 (2019). (Corrected version of [16].)

work page arXiv 1901
[18]

N., Murthy, M

Tran, M. N., Murthy, M. V. N. and Bhaduri, R. K.: On the quantum density of states and parti- tioning an integer.Annals of Physics311(1) (2004), 204–219. DOI: 10.1016/j.aop.2003.12.004

work page doi:10.1016/j.aop.2003.12.004 2004
[19]

C.: Squares: additive questions and partitions.Int

Vaughan, R. C.: Squares: additive questions and partitions.Int. J. Number Theory11(2015), 1367–1409

work page 2015
[20]

M.: Asymptotic partition formulae

Wright, E. M.: Asymptotic partition formulae. III. Partitions intok-th powers.Acta Math.63 (1934), no. 1, 143–191. Jos´e L. Fern´andez Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, Madrid, Spain joseluis.fernandez@uam.es V´ıctor J. Maci´a Departamento de An´ alisis Matem´ atico, Universidad de La Laguna, Tenerife, Spain victor.macia@ull.edu.es

work page 1934

[1] [1]

Math.125 (1997), no

B´ aez-Duarte, L.: Hardy–Ramanujan’s Asymptotic Formula for Partitions and the Central Limit Theorem.Adv. Math.125 (1997), no. 1, 114–120

work page 1997

[2] [2]

and Miniconi, M.: Une ´ etude asymptotique probabiliste des coefficients d’une s´ erie enti` ere.J

Candelpergher, B. and Miniconi, M.: Une ´ etude asymptotique probabiliste des coefficients d’une s´ erie enti` ere.J. Th´ eor. Nombres Bordeaux26(2014), no. 1, 45–67

work page 2014

[3] [3]

L., Fern´ andez, P

Cant´ on, A., Fern´ andez, J. L., Fern´ andez, P. and Maci´ a, V. J.: Khinchin families and Hayman class. Comput. Methods Funct. Theory21 (2021), no. 4, 851–904

work page 2021

[4] [4]

L., Fern´ andez, P

Cant´ on, A., Fern´ andez, J. L., Fern´ andez, P. and Maci´ a, V. J.: Khinchin families, set constructions, partitions and exponentials.Mediterr. J. Math.21(2024), article no. 39, 28 pp. 16 J. L. FERN ´ANDEZ AND V. J. MACI ´A

work page 2024

[5] [5]

and Maci´ a V

Cant´ on, A., Fern´ andez, J.L., Fern´ andez, P. and Maci´ a V. J.: Growth of power series with nonnegative coefficients, and moments of power series distributions.Publicacions Matem` atiques70, no 1, (2026), 161–201

work page 2026

[6] [6]

Fern´ andez, J. L. and Maci´ a, V. J.:Large powers asymptotics, Khinchin families and Lagrangian distributions, arXiv:2201.11746 (2022)

work page arXiv 2022

[7] [7]

Number Theory163(2016), 19–42

Gafni, A.: Power partitions.J. Number Theory163(2016), 19–42

work page 2016

[8] [8]

Hardy, G. H. and Ramanujan, S.: Asymptotic formulae in combinatory analysis.Proc. London Math. Soc.(2)17(1918), 75–115

work page 1918

[9] [9]

K.: A Generalisation of Stirling’s Formula.J

Hayman, W. K.: A Generalisation of Stirling’s Formula.J. Reine Angew. Math.196(1956), 67–95

work page 1956

[10] [10]

Ikeda, K.: An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modularj-function.arXiv preprintarXiv:2510.10598 (2025)

work page arXiv 2025

[11] [11]

J.:Tesis doctoral/Ph

Maci´ a, V. J.:Tesis doctoral/Ph. D. Thesis: The Theory of Khinchin Families, Universidad Aut´ onoma de Madrid (October 2024). Available at arXiv:2503.14157

work page arXiv 2024

[12] [12]

J.: Some Gaussianity criteria for Khinchin families.J

Maci´ a, V. J.: Some Gaussianity criteria for Khinchin families.J. Math. Anal. Appl.556(2025), 130239

work page 2025

[13] [13]

Murthy, M. V. N., Brack, M., Bhaduri, R. K., and Bartel, J.: Semiclassical analysis of dis- tinct square partitions.Physical Review E98(2018), 052131. DOI: 10.1103/PhysRevE.98.052131. arXiv:1808.05146

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physreve.98.052131 2018

[14] [14]

C.: Probability and entire functions

Rosenbloom, P. C.: Probability and entire functions. InStudies in Mathematical Analysis and Re- lated Topics. Essays in Honor of G. P´ olya, Stanford Univ. Press, Stanford, CA, 1962, pp. 325–332

work page 1962

[15] [15]

Roth, K. F. and Szekeres, G.: Some asymptotic formulae in the theory of partitions.The Quarterly Journal of Mathematics,5, Issue 1, (1954) pp. 241–259

work page 1954

[16] [16]

Tenenbaum, G., Wu, J., and Li, Y.-L.: Power partitions and saddle-point method.Journal of Number Theory,204(2019), 435–445

work page 2019

[17] [17]

and Li, Y.: Power partitions and saddle-point method.arXiv:1901.02234v4 (2019)

Tenenbaum, G., Wu, J. and Li, Y.: Power partitions and saddle-point method.arXiv:1901.02234v4 (2019). (Corrected version of [16].)

work page arXiv 1901

[18] [18]

N., Murthy, M

Tran, M. N., Murthy, M. V. N. and Bhaduri, R. K.: On the quantum density of states and parti- tioning an integer.Annals of Physics311(1) (2004), 204–219. DOI: 10.1016/j.aop.2003.12.004

work page doi:10.1016/j.aop.2003.12.004 2004

[19] [19]

C.: Squares: additive questions and partitions.Int

Vaughan, R. C.: Squares: additive questions and partitions.Int. J. Number Theory11(2015), 1367–1409

work page 2015

[20] [20]

M.: Asymptotic partition formulae

Wright, E. M.: Asymptotic partition formulae. III. Partitions intok-th powers.Acta Math.63 (1934), no. 1, 143–191. Jos´e L. Fern´andez Departamento de Matem´ aticas, Universidad Aut´ onoma de Madrid, Madrid, Spain joseluis.fernandez@uam.es V´ıctor J. Maci´a Departamento de An´ alisis Matem´ atico, Universidad de La Laguna, Tenerife, Spain victor.macia@ull.edu.es

work page 1934