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arxiv: 2607.01210 · v1 · pith:CTFGNUD4new · submitted 2026-07-01 · 🧮 math.NT

On a conjecture of Andrews and almost alternating sign patterns

Jayashree Kalita , Debanjana Kundu , Matthias Storzer , Xintong Wang This is my paper

Pith reviewed 2026-07-02 06:37 UTC · model grok-4.3

classification 🧮 math.NT
keywords q-seriesalternating signsRamanujan's Lost Notebookcircle methodq-hypergeometric seriesasymptoticssign patternsAndrews conjecture
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The pith

Coefficients of three q-series from Ramanujan's Lost Notebook alternate in sign except on a density-zero set of exceptions.

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

This paper proves that the coefficients of the q-series v2(q), v3(q), and v4(q) are alternating in sign, with only a density-zero set of exceptions. It derives precise asymptotic formulas via an adapted circle method, showing that the dominant term arises from the interplay between exponential growth and oscillatory behavior near roots of unity. The same sign behavior is established for explicit infinite families of q-hypergeometric series, where it arises systematically from those asymptotics. A sympathetic reader would care because this confirms a numerical observation first made by Andrews and indicates the pattern may be widespread among such series.

Core claim

For the q-series v2(q), v3(q), and v4(q), the coefficients are alternating in sign with only a density-zero set of exceptions; the same sign behavior holds for explicit infinite families of q-hypergeometric series and arises systematically from the oscillatory asymptotics of these series near roots of unity.

What carries the argument

Adapted circle method that yields asymptotic formulas whose dominant term is an alternating sign factor produced by exponential growth interacting with oscillatory behavior near roots of unity.

Load-bearing premise

Error terms in the asymptotic formulas do not disturb the sign pattern except on a density-zero set.

What would settle it

A proof or explicit computation showing that the exceptions to alternating signs have positive density for v2(q), v3(q), or v4(q).

Figures

Figures reproduced from arXiv: 2607.01210 by Debanjana Kundu, Jayashree Kalita, Matthias Storzer, Xintong Wang.

Figure 1
Figure 1. Figure 1: The coefficients V2(n) for 0 ≤ n ≤ 200. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The values (−1)n√ ne−2 √ nRe(√ W2)V2(n) for 0 ≤ n ≤ 500. Let j ∈ {2, 3, 4}. Following the philosophy of the circle method, the asymptotics of the coefficients Vj (n) can be obtained from the asymptotics of the generating function vj (q) for q near a root of unity. For example, when j = 2, the main contribution for v2(q) comes from q near the dominant pole −1 and we have v2(−e −z ) = "r π 4 √ 3z  e W2/z(1 … view at source ↗
Figure 3
Figure 3. Figure 3: The branch cuts of Liφ s (e −iv), with s = 1, 2 and −iφx + v for x ≥ 0. Following the standard construction of the dilogarithm, we define Liφ 2 for v ∈ C \ D by Liφ 2 (e −iv) := − Z e−iv 0 log(1 f − u) du u = (−φ) Z ∞ 0 logf [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The contours LR and CR. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The contours −izL∞ and S. In the integral representation (15) of Vr,b(−e −z ), we modify the contour −izL∞ to a contour S, going through the points −π/(6r) and π/(6r), cf [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The contour S = µ ∪ µ ′ (after applying Lemma 3.2). Upon taking derivatives, we see that f ′ r (v) = −rilog(1 − e −2riv) + 4v − sign(Re(v/φ))π, f ′′ r (v) = 2r 2 e 2irv − 1 + 4. (18) The critical points v0 of fr satisfy f ′ r (v0) = 0, and thus (1 − e −2riv0 ) r = −e −4iv0 . Equivalently, e −2riv0 = e ±πi/3 if r = 1 or r = 2. Given the shape of the contour and the branch cuts as in [PITH_FULL_IMAGE:figure… view at source ↗
Figure 7
Figure 7. Figure 7: The contour C and the major arc C ′ (where λ > 0). We now turn to the details of the proof. Proof of Theorem 5.1. Using Cauchy’s integral formula, we have V (n) = 1 2πi Z C v(q) q n+1 dq, where C is traversed exactly once in the counter-clockwise direction as seen in [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The contours Γ and Γ′ . We now expand the functions in the integrand about the saddle point √ W, expressing them as functions of w. We have √ ng(z) = √ n X∞ r=0 g (r) ( √ W) r! (iw) r n − r 4 = 2√ nW − W− 1 2 w 2 + X∞ r=3 (−i) r W 1 2 (1−r)w rn 2−r 4 , which implies e √ ng(z) = e 2 √ nW e −W− 1 2 w2  1 + O   X∞ r=1 n − r 4 per(w)     , (33) where each per(w) ∈ C[w]. Similarly, we have z − 1 2 = W… view at source ↗
Figure 10
Figure 10. Figure 10: Coefficients V {2} (n) for 0 ≤ n ≤ 800. The distinct colours also indicate an almost alternating sign pattern for the coefficients V {2} (n) in the sense of Theorem A. Moreover, the q-series satisfies the following relation with the Ramanujan’s third order mock theta function ν v {2} (q) = (q; q)∞ ν(q), which follows from the definition of both the q-series. Further it was shown in [17, Equation (2.6)] th… view at source ↗
Figure 11
Figure 11. Figure 11: The coefficients V {3} (n) for 0 ≤ n ≤ 200. References [1] R. Agarwal. On the paper “A ‘lost’ notebook of Ramanujan”. Adv. Math., 53(3):291–300, 1984. [2] G. E. Andrews. Ramanujan’s “lost” notebook. IV. Stacks and alternating parity in partitions. Adv. Math., 53(1):55–74, 1984. [3] G. E. Andrews. Questions and conjectures in partition theory. Amer. Math. Monthly, 93(9):708–711, 1986. [4] G. E. Andrews. Ra… view at source ↗
read the original abstract

In this paper, we prove a sign phenomenon first observed by Andrews for certain $q$-series from Ramanujan's Lost Notebook. For three of the series considered by Andrews, namely $v_2(q)$, $v_3(q)$, and $v_4(q)$, we show that the coefficients are alternating in sign, with only a density-zero set of exceptions. Our approach yields precise asymptotic formulas for the coefficients via an adapted circle method, inspired by the work of Folsom-Males-Rolen-Storzer on the $q$-series $v_1(q)$, revealing an interplay between exponential growth and oscillatory behaviour. This interaction produces a dominant alternating sign factor, which governs the sign regularity observed numerically by Andrews. More broadly, we establish the same sign behaviour for explicit infinite families of $q$-hypergeometric series encompassing these examples, and show that it arises systematically from oscillatory asymptotics of these $q$-series near roots of unity. We introduce an additional family whose coefficients appear to exhibit similar sign regularity, suggesting that this phenomenon is widespread and may point towards a deeper underlying theory.

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

0 major / 2 minor

Summary. The manuscript proves that the coefficients of the q-series v_2(q), v_3(q), and v_4(q) from Ramanujan's Lost Notebook are alternating in sign except for a density-zero set of exceptions. It derives precise asymptotic formulas via an adapted circle method, showing that an alternating sign factor arises from the interplay between exponential growth and oscillatory behavior near roots of unity. The result is extended to explicit infinite families of q-hypergeometric series, with the sign behavior attributed systematically to oscillatory asymptotics; an additional family is introduced with apparent similar sign regularity.

Significance. If the asymptotic main terms and error estimates hold, the work rigorously confirms Andrews' observed sign phenomenon and supplies an analytic mechanism for sign patterns in q-series. The uniformity of the argument over the stated families and the density-zero control on exceptions are strengths. The explicit main-term formulas obtained from the adapted circle method provide a concrete, falsifiable explanation that goes beyond numerical observation.

minor comments (2)
  1. [§1] §1: the precise formulation of the Andrews conjecture addressed here (including the specific Lost Notebook entries) should be stated explicitly before the main theorems.
  2. The dependence of the error term on the auxiliary parameters in the infinite families should be tracked explicitly in the statement of the uniformity result (currently only summarized in the abstract).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the detailed summary of its contributions, and the recommendation for minor revision. The report contains no enumerated major comments, so we have no specific points to address.

Circularity Check

0 steps flagged

Minor self-citation in circle-method adaptation; central asymptotics independent

full rationale

The derivation applies an adapted circle method (inspired by Folsom-Males-Rolen-Storzer on v1(q)) to obtain explicit asymptotic main terms for v2(q), v3(q), v4(q) and infinite families. The alternating sign factor emerges directly from the oscillatory contribution near roots of unity in the major-arc integrals; error bounds are shown o of the main term outside a density-zero set. The cited prior work supplies the analytic technique but does not encode the target sign claim or the new series. No fitted parameter is relabeled as prediction, no self-definitional loop, and the load-bearing estimates are external to the present paper's inputs. This yields a low but non-zero circularity score solely from the overlapping authorship on the method source.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the proof invokes standard properties of q-hypergeometric series and the circle method; no free parameters or invented entities are mentioned. The oscillatory asymptotics near roots of unity are treated as a derived feature rather than an added axiom.

axioms (2)
  • domain assumption The circle method can be adapted to extract precise asymptotics for the coefficients of these specific q-hypergeometric series.
    Abstract states the approach 'yields precise asymptotic formulas via an adapted circle method' without detailing the adaptation.
  • domain assumption The error terms in the asymptotic expansion do not affect the sign pattern outside a density-zero set.
    Required for the 'only a density-zero set of exceptions' claim but not justified in the abstract.

pith-pipeline@v0.9.1-grok · 5729 in / 1566 out tokens · 22723 ms · 2026-07-02T06:37:11.904203+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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

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