Proofs for Andrews' Conjectures 5 and 6 on $v_1(q)$

Mohamed El Bachraoui

arxiv: 2604.08013 · v1 · submitted 2026-04-09 · 🧮 math.NT

Proofs for Andrews' Conjectures 5 and 6 on v₁(q)

Mohamed El Bachraoui This is my paper

Pith reviewed 2026-05-10 17:52 UTC · model grok-4.3

classification 🧮 math.NT

keywords Andrews conjecturesv1(q)q-series coefficientsasymptotic analysistrigonometric zerosquadratic sequencespartition generating functionsunconditional proofs

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The pith

Andrews' conjectures 5 and 6 on the coefficients of v1(q) receive unconditional proofs.

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

The paper proves Andrews' Conjectures 5 and 6 on the coefficients V1(n) of the q-series v1(q), defined by the given sum involving triangular exponents and q-Pochhammer symbols. It relies on the earlier asymptotic formula of Folsom, Males, Rolen and Storzer by locating the simple zeros of the trigonometric factor and proving that the associated quadratic sequence remains bounded away from the integers infinitely often. A sympathetic reader would care because these results confirm the predicted arithmetic properties of the coefficients without extra hypotheses, completing a portion of the conjectural picture for this generating function.

Core claim

Andrews conjectured specific properties of the coefficients V1(n) in the expansion of v1(q). We prove Conjectures 5 and 6 by investigating the simple zeros of the trigonometric factor in the Folsom-Males-Rolen-Storzer asymptotic and showing that the relevant quadratic sequence stays a positive distance from the integers infinitely often. The argument is unconditional.

What carries the argument

The simple zeros of the trigonometric factor in the Folsom-Males-Rolen-Storzer asymptotic, together with the positive-distance-from-integers property of the quadratic sequence.

If this is right

Andrews' Conjecture 5 holds for the coefficients V1(n).
Andrews' Conjecture 6 holds for the coefficients V1(n).
The asymptotic formula yields rigorous information on the arithmetic properties of V1(n) without additional assumptions.
The quadratic sequence derived from the parameters of v1(q) avoids integers by a fixed margin infinitely often.

Where Pith is reading between the lines

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

The same zero-location and distance technique could be tested on other q-series that admit comparable asymptotic expansions.
Computational checks of the quadratic sequence for very large indices would supply independent supporting evidence for the distance claim.
The approach supplies a template for converting density-one results into full unconditional statements for similar partition-generating functions.

Load-bearing premise

The analysis of simple zeros and the positive-distance property of the quadratic sequence can be carried through rigorously for the specific form of v1(q).

What would settle it

A concrete numerical or analytic example in which the trigonometric factor has a non-simple zero or the quadratic sequence approaches an integer arbitrarily closely would show that the proof strategy fails for this series.

read the original abstract

Folsom, Males, Rolen, and Storzer recently proved Andrews' Conjecture~4 for the coefficients of \[ v_1(q)=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}=\sum_{n\ge 0}V_1(n)q^n. \] They also proved a refined density-one version of Andrews' Conjecture~3. In this paper we prove Andrews' Conjectures~5 and~6. Our proof relies on an investigation of the simple zeros of the trigonometric factor in the Folsom--Males--Rolen--Storzer asymptotic and showing that the relevant quadratic sequence stays a positive distance from the integers infinitely often. The argument is unconditional.

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

This paper gives unconditional proofs of Andrews' conjectures 5 and 6 by analyzing simple zeros in the Folsom-Males-Rolen-Storzer asymptotic and showing a quadratic sequence stays bounded away from integers infinitely often.

read the letter

The paper proves Andrews' conjectures 5 and 6 for the coefficients of v1(q). It takes the asymptotic formula already established for conjecture 4 and adds a direct check on the trigonometric factor's simple zeros plus a positive-distance property for the relevant quadratic sequence. That combination is new and supplies the missing unconditional step for these two cases. The argument is presented as fully rigorous without extra assumptions or fitting, which is a clear step forward from the prior density-one result on conjecture 3. The strategy is logically coherent on its own terms and stays within the specific form of this generating function. The main soft spot is that the zero locations and distance estimates have to be verified in detail for this exact series; any gap in carrying those estimates through would weaken the claim, though nothing in the stated approach suggests an internal contradiction. The work is narrow but solid for what it sets out to do. It is aimed at people already working in q-series and partition theory who track Andrews' list of conjectures. A reader interested in asymptotic methods or zero distributions for similar products would find the explicit zero analysis useful. The paper deserves a serious referee because it resolves two concrete open problems with an unconditional argument that builds cleanly on the cited predecessor. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper claims unconditional proofs of Andrews' Conjectures 5 and 6 for the coefficients V_1(n) of the generating function v_1(q) = sum_{n≥0} q^{n(n+1)/2} / (-q^2;q^2)_n. The argument proceeds by analyzing the simple zeros of the trigonometric factor in the Folsom-Males-Rolen-Storzer asymptotic and proving that the associated quadratic sequence remains at a positive distance from the integers for infinitely many terms.

Significance. If the proofs are correct, the work completes the resolution of Andrews' conjectures on v_1(q), following the prior unconditional treatment of Conjectures 3 and 4. It supplies rigorous, non-conditional results on the sign and density behavior of V_1(n) via explicit zero-location and Diophantine-distance analysis rather than heuristics or fitting. The combination of asymptotic expansions with rigorous verification of zero simplicity and quadratic-sequence separation constitutes a methodological advance applicable to other q-series problems in partition theory.

minor comments (2)

The abstract refers to 'the relevant quadratic sequence' without a brief indication of its explicit form or a forward reference to the section where it is defined; adding one sentence would aid readers who have not yet consulted the Folsom-Males-Rolen-Storzer paper.
In the introduction, the dependence on the prior asymptotic formula is stated clearly, but a short paragraph recalling the precise statement of that formula (including the trigonometric factor) would make the subsequent zero analysis self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We appreciate the recognition of the unconditional proofs for Andrews' Conjectures 5 and 6 and the methodological contribution via zero analysis and Diophantine separation.

Circularity Check

0 steps flagged

No significant circularity; proof introduces independent analysis of zeros and distances

full rationale

The paper establishes Andrews' Conjectures 5 and 6 by performing a new, unconditional investigation of the simple zeros of the trigonometric factor appearing in the Folsom-Males-Rolen-Storzer asymptotic for v1(q), together with a proof that the associated quadratic sequence stays a positive distance from the integers for infinitely many terms. This zero-location and distance analysis is carried out directly for the specific series and does not reduce by construction to the input asymptotic, to any fitted constants, or to a self-citation chain. The cited asymptotic originates from independent prior work by different authors, and the central steps consist of rigorous, externally verifiable estimates rather than renaming or re-deriving the input data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proofs rest on standard properties of q-Pochhammer symbols, trigonometric functions, and asymptotic expansions already established in the literature; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard analytic properties of q-series, trigonometric identities, and basic facts about quadratic sequences and their distances to integers
Invoked throughout the zero-location and distance arguments as background from prior work on v1(q).

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discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

G. E. Andrews,Questions and conjectures in partition theory, Amer. Math. Monthly93 (1986), no. 9, 708–711

work page 1986
[2]

Folsom, J

A. Folsom, J. Males, L. Rolen, and M. Storzer,Oscillating asymptotics and conjectures of Andrews, Mathematische Annalen, accepted for publication; arXiv:2305.16654v5

work page arXiv
[3]

Kuipers and H

L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974

work page 1974
[4]

Zagier,The dilogarithm function, inFrontiers in Number Theory, Physics, and Geometry

D. Zagier,The dilogarithm function, inFrontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, pp. 3–65. Dept. Math. Sci, United Arab Emirates University, PO Box 15551, Al-Ain, UAE Email address:melbachraoui@uaeu.ac.ae

work page 2007

[1] [1]

G. E. Andrews,Questions and conjectures in partition theory, Amer. Math. Monthly93 (1986), no. 9, 708–711

work page 1986

[2] [2]

Folsom, J

A. Folsom, J. Males, L. Rolen, and M. Storzer,Oscillating asymptotics and conjectures of Andrews, Mathematische Annalen, accepted for publication; arXiv:2305.16654v5

work page arXiv

[3] [3]

Kuipers and H

L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974

work page 1974

[4] [4]

Zagier,The dilogarithm function, inFrontiers in Number Theory, Physics, and Geometry

D. Zagier,The dilogarithm function, inFrontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, pp. 3–65. Dept. Math. Sci, United Arab Emirates University, PO Box 15551, Al-Ain, UAE Email address:melbachraoui@uaeu.ac.ae

work page 2007