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arxiv: 2602.18575 · v3 · pith:37XA7OSOnew · submitted 2026-02-20 · 🧮 math.PR · math.CV· math.NT

Power Partitions and Hayman Functions

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, within the probabilistic framework of Khinchin families, that the generating function $P_k$ of partitions into $k$-th powers is strongly Gaussian in the sense of B\'aez-Duarte, and even further that it is a Hayman function. Thus the Hardy--Ramanujan asymptotic formula for the number $p_k(n)$ of partitions of $n$ into $k$-th powers which reads \[ p_k(n) \sim \frac{\alpha_k}{n^{(3k+1)/(2k+2)}} \exp\!\Big(\beta_k\, n^{1/(k+1)}\Big), \qquad n\to\infty, \] where $\alpha_k$ and~$\beta_k$ are explicit constants depending only on $k$, follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity of $P_k$ combines a Gaussianity criterion for Khinchin families with certain 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. Analogous results are presented for the generating function $Q_k$ of partitions into distinct $k$-th powers.

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.

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)
  1. [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.
  2. [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)
  1. [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.
  2. [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

  1. 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

  2. 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.

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