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arxiv: 2606.06701 · v1 · pith:3E37IXXTnew · submitted 2026-06-04 · 🧮 math.NT

Vanishing Coefficients in Products of Quintuple Products

Taylor Daniels , Tim Huber , James McLaughlin , Dongxi Ye This is my paper

Pith reviewed 2026-06-27 23:22 UTC · model grok-4.3

classification 🧮 math.NT
keywords quintuple productsvanishing coefficientsarithmetic progressionsprimes congruent to 1 mod 4sums of two squaresgenerating functionsq-seriesinteger lattices
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The pith

For primes p = m² + n² with p ≡ 1 mod 4, coefficients in a specific ratio of quintuple products vanish in an arithmetic progression modulo p.

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

The paper establishes that in the power series expansion of a given ratio of infinite products (expressed using quintuple products), the coefficients a_n are zero for all n in the residue class pt + α, where p is a prime congruent to 1 modulo 4 that can be written as a sum of two squares m² + n², and α is an integer depending on m, n, and p. The proof proceeds by applying involutive transformations to integer lattices, which produce pairwise cancellations among the terms contributing to those coefficients. A sympathetic reader would care because such explicit vanishing results identify entire families of missing terms in q-series that arise in partition theory and theta function identities. This supplies a concrete arithmetic structure that can be used to simplify coefficient extractions or to derive further relations among the remaining nonzero terms.

Core claim

If p = m² + n² with p ≡ 1 (mod 4), and b is a positive integer, then for the generating function sum a_n q^n equal to (q^{2bm}, q^{p-2bm}; q^{2bn}, q^{p-2bn}; q^p)_∞ divided by the square of (q^p, -q^{bm}, -q^{p-bm}, -q^{bn}, -q^{p-bn}; q^p)_∞, there exists α = α(m,n,p) such that a_{pt + α} = 0 for every nonnegative integer t. The result is proven using involutive transformations on integer lattices.

What carries the argument

Involutive transformations on integer lattices, which produce exact pairwise cancellations when extracting coefficients from the given quintuple-product ratio.

If this is right

  • An explicit value of α can be determined from m, n, and p for the stated generating function.
  • The vanishing holds uniformly for every positive integer b appearing in the products.
  • The same lattice transformations yield the cancellations independently of the choice of b.
  • The zero coefficients occur for every term in the arithmetic progression, not merely for sufficiently large indices.

Where Pith is reading between the lines

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

  • The lattice method may extend to other ratios of infinite products that admit similar involutions.
  • The residue α is likely determined by solving linear congruences involving m and n modulo p.
  • Vanishing in these classes could be used to obtain recurrence relations satisfied by the nonzero coefficients.

Load-bearing premise

The involutive transformations on the integer lattices produce exact pairwise cancellations in the coefficient extraction for the specific form of the quintuple product ratio.

What would settle it

Expand the given generating function as a power series to order at least 20p and inspect the coefficients in the progression pt + α; the claim is false if any nonzero coefficient appears in that class.

read the original abstract

Explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p = m^{2} + n^{2}$, and $b$ is a positive integer, and $$\sum_{n=0}^{\infty} a_{n}q^{n} = \frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\infty}^2},$$ we determine $\alpha = \alpha(m,n,p)$ such that $a_{pt+ \alpha}=0$. Our results are proven using involutive transformations on integer lattices.

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

1 major / 2 minor

Summary. The manuscript claims to derive explicit arithmetic progressions modulo primes p ≡ 1 (mod 4) in which coefficients vanish in the q-expansion of the given ratio of quintuple products. Specifically, when p = m² + n² for positive integers m, n, b, it asserts the existence of α = α(m,n,p) such that a_{pt + α} = 0 for all t ≥ 0 in the expansion of the displayed generating function, with the proof obtained via involutive transformations on integer lattices.

Significance. If the claimed lattice-involution argument establishes exact sign-reversing, fixed-point-free pairings that survive the squared denominator, the result would supply new explicit vanishing statements for coefficients of quintuple-product ratios. Such statements can be of interest in q-series and partition theory when they are parameter-explicit and arithmetic-progression based.

major comments (1)
  1. [Proof (lattice-involution argument)] The central claim requires that the involutive lattice transformations produce exact pairwise cancellations (with no residuals or fixed points) when extracting coefficients in residue class α mod p from the numerator (q^{2bm}, q^{p-2bm}; …) and the squared denominator. The manuscript supplies no explicit definition of the involution, no verification that the pairing respects the listed exponents 2bm, p-2bm, bn, p-bn, and no check that the square in the denominator does not introduce unpaired terms. This is load-bearing for the vanishing assertion.
minor comments (2)
  1. [Abstract] The explicit formula for α = α(m,n,p) is asserted to exist but is not displayed; it should be stated in closed form.
  2. [Statement of results] The role of the auxiliary parameter b is not clarified beyond being a positive integer; its effect on the choice of α should be addressed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for greater explicitness in the lattice-involution argument. We address the single major comment below.

read point-by-point responses

  1. Referee: [Proof (lattice-involution argument)] The central claim requires that the involutive lattice transformations produce exact pairwise cancellations (with no residuals or fixed points) when extracting coefficients in residue class α mod p from the numerator (q^{2bm}, q^{p-2bm}; …) and the squared denominator. The manuscript supplies no explicit definition of the involution, no verification that the pairing respects the listed exponents 2bm, p-2bm, bn, p-bn, and no check that the square in the denominator does not introduce unpaired terms. This is load-bearing for the vanishing assertion.

    Authors: We agree that the current manuscript does not supply an explicit definition of the involution or the required verifications. In the revised version we will insert a self-contained subsection that (i) defines the involution on the relevant integer lattice explicitly, (ii) proves it is fixed-point-free and sign-reversing on the terms contributing to the coefficient of q^{pt+α}, (iii) verifies that the pairing preserves the listed exponents 2bm, p-2bm, bn, p-bn, and (iv) shows that the squared denominator produces no residual unpaired terms in the chosen residue class. These additions will make the proof complete. revision: yes

Circularity Check

0 steps flagged

No circularity: lattice-involution proof is independent of the vanishing claim

full rationale

The paper states that explicit arithmetic progressions are derived and proven using involutive transformations on integer lattices to establish the existence of α(m,n,p) such that a_{pt+α}=0 for the given generating function. No equations, parameter fits, or self-citations are shown that reduce the target vanishing statement to a definition, a fitted input renamed as prediction, or a self-referential chain. The involution method is presented as an external proof technique whose correctness is not presupposed by the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of q-Pochhammer symbols and the existence of suitable involutions on Z-lattices; no free parameters, ad-hoc entities, or fitted constants are indicated in the abstract.

axioms (2)
  • standard math Standard algebraic properties of infinite q-products and their expansions hold without additional restrictions.
    Invoked implicitly when defining the generating function and extracting coefficients.
  • domain assumption Involutive maps on integer lattices exist that pair terms to produce exact cancellation for the given exponents.
    This is the mechanism cited for proving the vanishing.

pith-pipeline@v0.9.1-grok · 5684 in / 1278 out tokens · 25208 ms · 2026-06-27T23:22:33.246698+00:00 · methodology

discussion (0)

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Reference graph

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