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Sign-patterns of Certain Infinite Products

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Pith's one-line read For every prime p>3 and integer i>1 not divisible by p, the coefficients of (q^i;q^i)_∞/(q^p;q^p)_∞ are eventually periodic in sign modulo p, with an explicit threshold; the case (p,i)=(5,2) proves a conjecture from [9].

desk verdict Genuinely new sign-pattern results that resolve Bringmann et al. conjectures, but the central dissection formula is imported from the authors' unpublished preprint and needs to be proved in the paper. read the letter →

arxiv 2507.16644 v1 pith:DZJNZ2KK submitted 2025-07-22 math.NT

Zeyu Huang , Timothy Huber , James McLaughlin , Pengjun Wang , Yan Xu , Dongxi Ye This is my paper · ORCID
classification math.NT MSC 11F3030C50
keywords vanishingcoefficientsperiodicsignchangesofinfiniteq-productsetaquotientsquintupleproductidentitym-dissectionsFourier
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that the signs of the coefficients in the series expansion of $(q^i;q^i)_\infty/(q^p;q^p)_\infty$ are eventually periodic: for any prime $p>3$ and integer $i>1$ not divisible by $p$, once $n$ exceeds an explicit bound $N(p,i)$, the sign of $a_n$ is determined entirely by $n$ modulo $p$. In residue classes that can be written as $i(6r^2+r) \pmod p$, the coefficient is positive for small and large $r$ and negative for the middle range of $r$; in every other residue class it vanishes. The thresholds that split these ranges depend only on whether $p$ is $1$ or $-1$ modulo $3$. Specializing to $(p,i)=(5,2)$ confirms the conjecture $C215^{-1}$ from [9] for all $n \ge 0$. The same dissection technique settles several further sign and vanishing conjectures from [9] for related eta quotients, including $(q;q)^9/(q^3;q^3)^9$.

What carries the argument

The carrying mechanism is the $m$-dissection formula for the quintuple product (Theorem 2.1), taken from [22]. It splits the product $(q^j,q^{M-j},q^M;q^M)_\infty(q^{M-2j},q^{M+2j};q^{2M})_\infty$ into a finite signed sum with exponents $L(r)=6r^2+\cdots$ and with each summand a product whose ordinary power series has nonnegative coefficients. Setting $M=4$, $j=1$ turns this into a $p$-dissection of $(q;q)_\infty$ (Corollary 2.1), in which the sign of the $r$-th summand is $(-1)^{s(r)}$ and $s(r)$ takes the values $0,1,2$ according to the same thresholds on $r$. After replacing $q$ by $q^i$ and dividing by $(q^p;q^p)_\infty$, only summands whose exponent $iL(r)$ lies in a given residue class contribute to $a_n$, and Lemma 2.1 shows those contributions are strictly positive or negative eventually. Consistency of the sign within each residue class follows from the symmetry of the quadratic $6r^2+r$ modulo $p$: for $p\equiv 1 \pmod 3$, the two outer ranges of $r$ are interchanged by $r+r'\equiv (p-1)/6 \pmod p$, and the case $p\equiv -1 \pmod 3$ is analogous.

What would settle it

Expand both sides of Theorem 2.1 for $M=4$, $j=1$, and $m=5$ as power series in $q$ to order $q^{60}$; if the sides disagree, the load-bearing dissection is false. Equivalently, directly expand $(q^2;q^2)_\infty/(q^5;q^5)_\infty$: a nonzero coefficient with $n\equiv 1$ or $3 \pmod 5$, or a coefficient that is not positive when $n\equiv 0 \pmod 5$, would disprove the theorem's corollary.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1.1. Let $f_j=(q^j;q^j)_\infty$ and write $f_i/f_p=\sum_{n=0}^\infty a_n q^n$. For $p\equiv 1 \pmod 3$, define $L(r)$ piecewise with breakpoints $(4p-1)/12$ and $(10p-1)/12$; for $p\equiv -1 \pmod 3$, the breakpoints are $(2p-1)/12$ and $(8p-1)/12$. Then for every $n>N(p,i)$, where $N(p,i)$ is the maximum, over residue classes $s \pmod p$, of the least value of $iL(r)$ congruent to $s$, the paper proves $a_n>0$ when $n\equiv i(6r^2+r) \pmod p$ with $r$ in the outer intervals, $a_n<0$ in the inner interval, and $a_n=0$ otherwise. Corollary 1.1 is the case $(p,i)=(5,2)$: the coefficients of $(q^2;q^2)_\infty/(q^5;q^5)_\infty$ are positive exactly for $n\equiv 0 \pmod 5$, negative for $n\equiv 2,4 \pmod 5$, and zero for $n\equiv 1,3 \pmod 5$, for every $n\ge 0$. Theorem 1.2 extends the same sign analysis to other eta quotients, including $(q;q)^9/(q^3;q^3)^9$ with period-9 signs and several cases in the table from [9].

Load-bearing premise

Everything rests on an unproved dissection identity quoted from the authors' earlier preprint [22]: the quintuple product can be split into a signed finite sum whose pieces have nonnegative coefficients, and if that identity is false, the $p$-dissection of $(q;q)_\infty$ and the resulting sign patterns do not follow.

Editorial extensions

If this is right

  • For every prime $p>3$ and every integer $i>1$ not divisible by $p$, the coefficients of $(q^i;q^i)_\infty/(q^p;q^p)_\infty$ are ultimately periodic in sign with period $p$, and the threshold $N(p,i)$ is explicit.
  • The case $(p,i)=(5,2)$ settles the conjecture $C215^{-1}$ from [9] in full: $a_n>0$ for $n\equiv 0 \pmod 5$, $a_n<0$ for $n\equiv 2,4 \pmod 5$, and $a_n=0$ otherwise.
  • Theorem 1.2 proves the specific sign patterns listed for the entries $5/++0+0$, $4/+-00$, $3/+0-$, $4/+-+0$, $4/+++0$, $5/+- - ++$, and $9/+-+--+0-+$ in the table from [9].
  • Whenever $\gcd(t,6)=1$, Corollary 2.1 yields a $t$-dissection of $(q;q)_\infty$, so the same method can be applied to quotients such as $(q^j;q^j)^m/(q^t;q^t)$ and $(q^j;q^j)^m/(q^t;q^t)^m$, although the dissections may then be sums of products with mixed signs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An unstated consequence is that the proof strategy is algorithmic: for any fixed $(p,i)$, the bound $N(p,i)$ is computable, so the eventual sign pattern can be verified by checking only finitely many coefficients.
  • A testable extension of the same dissection idea would be to rational powers of the product $(q;q)_\infty/(q^p;q^p)_\infty$; numerical experiments for small $p$ could reveal which residue classes remain single-signed and which develop mixed signs.
  • The paper's own table of coefficient counts for two unproven quotients suggests that for some residue classes the coefficients may never become single-signed; if that persists, an eventual sign-periodicity statement for those classes would need a weaker formulation than pure $+$ or $-$.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the sign patterns of the Fourier coefficients of the eta quotient (q^i;q^i)_∞/(q^p;q^p)_∞ for primes p>3 and integers i>1 with p∤i. Theorem 1.1 asserts that, for n larger than an explicit bound N(p,i), the sign of the coefficient an is determined entirely by n mod p: the positive, negative, and zero residues are specified in terms of the quadratic L(r)=6r^2+r and the residue of p modulo 3. Corollary 1.1 specializes this to (p,i)=(5,2) and thereby proves a conjecture of Bringmann et al. (C215^-1). Theorem 1.2 gives sign-periodicity results for ten further eta quotients, several of which address other conjectures from the same source. The proofs are based on m-dissections, a quintuple-product dissection formula, theta-function identifications, and a lemma asserting that certain quotients of infinite products have nonnegative coefficients.

Significance. If the main results are correct, the paper provides a uniform method for determining sign patterns of a natural family of eta quotients and proves several open conjectures from the Bringmann et al. program, including the previously conjectured sign pattern for (q^2;q^2)_∞/(q^5;q^5)_∞. The explicit N(p,i) bounds and the sharpness table are also useful. However, the central theorem depends on an unproved m-dissection formula quoted from the authors' own unpublished preprint, and the statement of N(p,i) is formally ill-defined for some (p,i); these issues must be addressed before the results can be considered fully established. The paper does not provide machine-checked proofs, but the small cases (p,i)=(5,2) and (7,2) are consistent with the stated patterns.

major comments (3)
  1. [Section 2.1, Theorem 2.1] Theorem 2.1, the m-dissection formula for the quintuple product, is quoted from reference [22], an unpublished preprint by three of the authors, and is not proved in this manuscript. This formula is the sole source of Corollary 2.1, which is then used in every step of the proof of Theorem 1.1 in Section 3. Since a sign or exponent error in s(r) or L(r) would change the claimed positive/negative/zero classification, the main theorem is currently contingent on an unverified black box. The authors should either provide a complete proof of Theorem 2.1 in an appendix or replace [22] with a published source that the reader can check.
  2. [Theorem 1.1, definition of N] The definition N = max(∪_{s=0}^{p-1} min{iL(r) : iL(r) ≡ s (mod p), 0 ≤ r ≤ p−1}) − p is not well-formed, because for some (p,i) the set over which min is taken is empty. For example, when (p,i)=(5,2), the residues 1 and 3 modulo 5 are not represented by iL(r) for any r, so min over an empty set is undefined and N is undefined. The statement can be repaired by taking the maximum only over nonempty residue classes (or by defining min(∅)=+∞), but as written the theorem is not formally meaningful for all admissible (p,i).
  3. [Section 3, proof of Theorem 1.1] The proof asserts without detail that the quantities t1(r), 4p^2−t1(r), 4p^2, 4p^2+t1(r), 8p^2−t1(r), 8p^2, t2(r), and 8p^2−t2(r) are pairwise distinct and all multiples of p. This verification is load-bearing: it is exactly what ensures that after division by (q^p;q^p)_∞ each numerator factor (1−q^{iα}) cancels a distinct factor of the denominator, so that the remaining series has nonnegative coefficients and the sign of each term is governed by (−1)^{s(r)}. The authors should provide the omitted verification for both cases p≡1 (mod 3) and p≡−1 (mod 3), including a demonstration that no overlap occurs between the arithmetic progressions generated by the different factors.
minor comments (4)
  1. [Title and Abstract] The title contains a spacing artifact ('P A TTERNS') that should be corrected, and the abstract can be made more precise about the scope of the 'additional classes' covered by Theorem 1.2.
  2. [References] Reference [3] contains a typo, 'Ramunujan' instead of 'Ramanujan'; reference [22] is an unpublished preprint whose status should be updated if it has appeared.
  3. [Section 4, proof of Theorem 1.2(5)] In the proof of part (5), the congruence argument 'if n1^2+m1^2 ≡ n2^2+m2^2 (mod 4), then (n1+n2)(n1−n2) ≡ (m1+m2)(m1−m2) (mod 2)' is not written correctly; the left- and right-hand sides should both involve differences (n1−n2) and (m1−m2). The intended conclusion about parity is likely correct, but the displayed line needs correction.
  4. [Theorem 1.2(2)] Part (2) states 'For p = 1 or an odd prime'; the case p=1 should be separated from the prime case for clarity, since p=1 is not prime.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main sign pattern is derived from a general dissection identity, and the self-cited Theorem 2.1 is parameter-free and independent of the target result.

full rationale

Theorem 1.1 is not assumed anywhere in the paper; it is deduced from a p-dissection of (q;q)_infinity obtained in Corollary 2.1 by substituting M=4, j=1 into Theorem 2.1. The sign pattern emerges only after specializing that dissection and applying Lemma 2.1 to products with nonnegative coefficients, so the conclusion is not equivalent by construction to any input. The main provenance concern is that Theorem 2.1 is quoted from the authors' unpublished preprint [22] and not proved in the paper; however, this is a general quintuple-product dissection identity whose assumptions do not include the target sign pattern and which contains no fitted parameters, so under the stated rules it is independent support and does not constitute circularity. The remaining derivations rest on standard external identities (Borwein cubic theta functions, Ramanujan dissections, Cohen-Stromberg theta identities) and elementary coefficient-positivity lemmas. The formally problematic definition of N(p,i) using minima over possibly empty residue classes in Theorem 1.1 is a well-definedness and correctness issue, not a circularity. Overall, no equation in the paper reduces to its own input by construction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests mainly on the unpublished dissection formula (Theorem 2.1) from the authors' own preprint, plus standard theta identities and a set of 'routine' verifications that are asserted but not shown. No fitted parameters or invented entities are introduced.

assumptions (5)
  • ad hoc to paper Theorem 2.1: m-dissection formula for the quintuple product, quoted from [22]
    This is the central tool. It is stated without proof and taken from the authors' unpublished preprint. The proof of Theorem 1.1 depends on it via Corollary 2.1.
  • standard math Lemma 2.1: quotient of products expands with nonnegative coefficients
    The reciprocal of an infinite product (1-q^n) expands into a series with nonnegative coefficients; used throughout to propagate signs.
  • standard math Jacobi triple product and theta identities (2.6)-(2.9)
    Identities from Cohen-Stromberg and Cooper used to convert products into theta series for Theorem 1.2.
  • standard math Borwein cubic theta function identities (2.4) and Ramanujan's 5-dissection of (q;q)
    Cited from Cooper [16] and Ramanujan [19]; used in Lemma 2.2 and Theorem 1.2(7).
  • domain assumption Pairwise distinctness of exponents t1(r), t2(r), 4p^2-t1(r), 8p^2-t2(r), etc. within each dissection term
    The proof asserts this is 'routine to check' but does not demonstrate it. If two exponents coincided, an uncanceled (1-q^e) factor could inject negative coefficients and break the sign argument.

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Pith. "Pith review of Sign-patterns of Certain Infinite Products." pith.science (2026). https://pith.science/paper/DZJNZ2KK

@misc{pith2026250716644,
  author       = {Pith},
  title        = {Pith review of: Sign-patterns of Certain Infinite Products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZJNZ2KK}},
  note         = {Machine review of arXiv:2507.16644}
}
abstract

The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for \[ \frac{(q^i;q^i)_{\infty}}{(q^p;q^p)_{\infty}} \] for integers \( i > 1 \) and primes \( p > 3 \). The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of \( (q^2;q^2)_{\infty}(q^5;q^5)_{\infty}^{-1} \). The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.

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