REVIEW 1 major objections 4 minor 8 references
Proofs for certain Pi_{q}-conjectures of Gosper
T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper proves that the 2001 conjectured identities for the q-constant $\Pi_q$ are theorems, deriving them from modular equations of degrees 3 and 5.
desk verdict A useful, mostly sound paper that turns several of Gosper's Pi_q conjectures into theorems via classical modular equations, with one clear misprint in equation (2.20) that needs fixing before publication. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is a modular equation of degree $n$: a relation between $\alpha=k^2$ and $\beta=\ell^2$ forced by comparing hypergeometric $\,_2F_1$ ratios, together with the multiplier $m=z_1/z_n$ where $z_n=\varphi(q^n)^2$. The paper proves two auxiliary theorems, one for $n=3$ and one for $n=5$, giving new algebraic identities in $\alpha,\beta,m$. Standard transcription formulas express $\psi(q^k)$ as $\sqrt{z_k}/2$ times a power of $\alpha/q$ or $\beta/q^k$; substituting these into $\Pi_q=q^{1/4}\psi(q)^2$ turns every conjectured identity into one of the algebraic statements just derived. The multiplier $m$ is the quantity that makes all the powers of $q$ come out correctly.
What would settle it
Evaluate both sides of identity (2.17), the $\psi$-form of (2.1), at $q=1/2$ and $q=1/3$ using $\psi(q)=\sum_{n\ge0}q^{n(n+1)/2}$ with 50-digit precision; any nonzero difference refutes Theorem 2.1, because the modular-equation derivation forces equality for $0<q<1$. The same test applied to (3.25) at $q=1/2$ checks the degree-5 family.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is that the conjectured identities (1.4)-(1.6), (2.1)-(2.5), and (3.1)-(3.5) are true. These are exact algebraic identities for $\Pi_q$ at scaled arguments: the first group involves $\Pi_q,\Pi_{q^2},\Pi_{q^3},\Pi_{q^6}$ and the second involves $\Pi_q,\Pi_{q^2},\Pi_{q^5},\Pi_{q^{10}}$. Writing $\psi(q)=\sum_{n\ge 0}q^{n(n+1)/2}$ and using $\Pi_q=q^{1/4}\psi(q)^2$, each identity is translated into a relation among ratios of $\psi$ values. Those relations are then shown to follow from modular equations of degree 3 (Theorem 2.2) and degree 5 (Theorem 3.2), with the classical multiplier $m$ carrying the algebra. The proof is completed by analytic continuation from $0<q<1$ to the full unit disk.
Load-bearing premise
The load-bearing assumption is that the standard transcription formulas for $\psi(q^k)$ in terms of $\alpha,\beta,z_1,z_n$ are valid simultaneously with the multiplier $m$, and that the identities proved for $0<q<1$ extend to $|q|<1$ with consistent branches for fractional powers such as $q^{1/4}$.
Editorial extensions
If this is right
- The two families of $\Pi_q$ identities are no longer conjectural; each is a theorem derived from a modular equation, valid for every $|q|<1$.
- Because (1.5) and (1.6) are equivalent, proving one of them proves both, so the paper's framework compresses the list of independent checks.
- The auxiliary modular equations (Theorems 2.2 and 3.2) are stated in a reusable form: any future $\Pi_q$ identity that reduces to the same algebraic relations is settled by the same proof.
- Each identity yields polynomial relations among the $\psi$-values $\psi(q^k)$, which can be verified numerically at any $q$ in $(0,1)$ to machine precision.
Reading between the lines
- The same template should extend to other prime degrees: deriving degree-7 modular equations and their multiplier formulas would predict a matching family of identities linking $\Pi_q,\Pi_{q^2},\Pi_{q^7},\Pi_{q^{14}}$.
- The paper leaves branches of $q^{1/4}$ implicit in the continuation step; a numerical check on the negative real axis, using a principal branch, would test whether the analytic continuation was made consistent.
- The algebraic relations suggest that the ratios $\Pi_{q^k}/\Pi_{q^l}$ satisfy a lattice of polynomial identities; mapping that lattice could uncover additional identities beyond the printed list.
- Since the proofs are entirely substitution-based, a reader could restate each identity as an identity of modular forms; that restatement might reveal why degree 3 and degree 5 are exactly the degrees needed for these two families.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves a batch of conjectured identities for Gosper's q-constant Pi_q, using classical modular equations of degrees 3 and 5. After translating each Pi_q identity into an equivalent identity for Jacobi's theta/psi functions, the author derives these equivalents from modular equations taken from Berndt's texts, with several auxiliary modular-equation identities proved in Theorems 2.2 and 3.2. Theorems 2.1 and 3.1 then assert the conjectured identities (1.4)--(1.6), (2.1)--(2.5), and (3.1)--(3.5). The proofs are algebraic and explicit, with the degree-5 section relying on large but explicitly stated polynomial reductions. The paper also uses a known equivalence of El Bachraoui to reduce the pair (1.5),(1.6) to a single proof.
Significance. The identities in question originated as empirical conjectures in Gosper's work on q-trigonometry, and a proof that settles them is a useful contribution to the subject. The paper's method is sound in conception: it reduces the q-constant identities to classical modular equations, avoids fitting or circular reasoning, and gives enough detail that the main deductions are checkable. A notable strength is that the degree-3 and degree-5 auxiliary identities are stated as explicit formulas with derivations indicated from Berndt's modular equations. However, the manuscript contains one false displayed equation, Eq. (2.20), which invalidates the proof of identity (2.4) as written; the defect is a repairable typo, but it must be corrected before the paper can be accepted.
major comments (1)
- [Section 2.3, Eq. (2.20)] The denominator on the right-hand side of (2.20) is misprinted: it should be ψ^4(q), not ψ^2(q). Dividing (2.11) by q^{1/2} and using (2.26) and (2.28) gives (1/m)(β/α)^{1/4} = q^{1/2} ψ^2(q^3)/ψ^2(q) and α^{1/2}/4 = q^{1/2} ψ^4(q^2)/ψ^4(q). The right-hand side after the division is therefore ψ^4(q^2)/ψ^4(q)(1 ∓ q^{1/2}ψ^2(q^3)/ψ^2(q))(1 ± 3q^{1/2}ψ^2(q^3)/ψ^2(q))^3, not ψ^4(q^2)/ψ^2(q) times the same bracket. As printed, (2.20) is false: for q=1/2 the left side is about 1.885, while the printed right side is about 5.08; the corrected denominator gives agreement. Since (2.20) is the claimed ψ-function equivalent of Gosper's identity (2.4), the proof of Theorem 2.1 for (2.4) as written is invalid. The correction is local and the derivation described in the text then goes through, but the manuscript must be amended.
minor comments (4)
- [Sections 2 and 3, analytic continuation] The statements 'By analytic continuation, these identities are also true for |q|<1' require a branch specification for q^{1/4} and related fractional powers. The identities are derived for 0<q<1; to conclude the full unit disk case, the author should state the chosen branch (e.g., the principal branch on the unit disk cut along the negative real axis) and note that the final equations are algebraic in the branch, so the continuation is legitimate.
- [Section 3.2, proof of Theorem 3.2] The proof of (3.7)--(3.10) asserts, after substitution and simplification, that both sides reduce to the listed polynomials A(m), B(m), C(m), and D(m). The intermediate algebra is not shown. For a formal publication, please provide the expansion or a verifiable certificate (e.g., a CAS output or a supplementary computation) so that the reader does not have to rerun a lengthy symbolic calculation.
- [Section 2.3, proof of (2.17)] The multiplier described before the proof of (2.17) is typeset in a garbled way: '(αβ)^{1/8}√(z1z3/q^3 z_3^2)/32' is not a readable formula. Please rewrite the factor unambiguously.
- [References] Reference [6] is listed with a URL and no volume or page numbers; please supply the complete bibliographic data or a DOI once available.
Circularity Check
No circularity: the proof chain reduces Gosper's Pi_q conjectures to independent classical modular equations from Berndt.
full rationale
The paper's central claims, Theorems 2.1 and 3.1, are proved by converting Pi_q identities into theta-function identities via the definition Pi_q = q^{1/4} psi(q)^2, then using standard modular equation relations taken from Berndt's books ([2] and [3]). These modular equations are external, parameter-free inputs; they are not defined in terms of the Pi_q identities being proved, nor are any parameters fitted to a subset of the target data. The only self-citation is reference [6], which supports identity (1.3), and (1.3) is not used in the derivations of Theorems 2.1 or 3.1. The analytic-continuation step from 0<q<1 to |q|<1 is a standard independent argument. Potential concerns such as the apparent typographical error in equation (2.20) are matters of correctness or presentation, not circularity: fixing or rejecting that step does not change the fact that the derivation relies on external modular equations rather than on its own conclusions. No load-bearing reduction to inputs by construction, and no renaming of a known result as a prediction, appears in the paper.
Assumptions & free parameters
assumptions (2)
- standard math Classical modular equation and theta function formulas from Berndt [2] hold with the stated multiplier m.
- domain assumption The identities extend from real q in (0,1) to complex q with |q|<1 by analytic continuation with consistent branches for fractional powers.
Cite this review
Pith. "Pith review of Proofs for certain Pi_{q}-conjectures of Gosper." pith.science (2026). https://pith.science/paper/DMEWXF4I
@misc{pith2026190802010,
author = {Pith},
title = {Pith review of: Proofs for certain Pi_q-conjectures of Gosper},
year = {2026},
howpublished = {\url{https://pith.science/paper/DMEWXF4I}},
note = {Machine review of arXiv:1908.02010}
}
read the original abstract
In 2001 W. Gosper introduced a constant Pi_{q} and conjectured without proofs many intriguing identities on this constant. In this paper we establish some modular equations of degrees 3 and 5. From these modular equations we confirm two groups of Pi_{q}-identities in Gosper's list. One group involves Pi_{q},Pi_{q^{2}},Pi_{q^{3}} or Pi_{q^{6}} while the other is related to Pi_{q},Pi_{q^{2}},Pi_{q^{5}} or Pi_{q^{10}}.
Reference graph
Works this paper leans on
-
[1]
S. Abo Touk, Z. Al Houchan and M. El Bachraoui, Proofs for t wo q-trigonometric identities of Gosper. J. Math. Anal. Appl. 456(1)(2017), 662–670
work page 2017
-
[2]
B.C. Berndt, Number Theory in the Spirit of Ramanujan, Am erican Mathematical Society, Providence, RI, 2006
work page 2006
-
[3]
Berndt, Ramanujan’s Notebooks, Part III, Springer -Verlag, New York, 1991
B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer -Verlag, New York, 1991
work page 1991
-
[4]
El Bachraoui, On the Gosper’s q-constant Π q
M. El Bachraoui, On the Gosper’s q-constant Π q. Acta Mathematica Sinica, English Series. 34(11)(2018), 1755–1764
work page 2018
-
[5]
Gosper, Experiments and discoveries in q-trigonometry, in: F.G
R.W. Gosper, Experiments and discoveries in q-trigonometry, in: F.G. Garvan, M.E.H. Ismail (Eds.), Symbolic Computation, Number Theory, Special Func tions, Physics and Combina- torics, Kluwer, Dordrecht, Netherlands, 2001, pp.79–105
2001
-
[6]
B. He and H.-C. Zhai, Proofs for certain q-trigonometric identities of Gosper, Science China–Mathematics, 2020, http://engine.scichina.com/d oi/10.1007/s11425-019-9555-1
-
[7]
Z.-G. Liu, Addition formulas for Jacobi theta functions , Dedekind’s eta functions, and Ra- manujan’s congruences, Pacific J. Math. 240(1)(2009), 135– 150
work page 2009
-
[8]
Liu, An addition formula for the Jacobian theta fun ction and its applications, Adv
Z.-G. Liu, An addition formula for the Jacobian theta fun ction and its applications, Adv. Math. 212(1)(2007), 389–406. School of Mathematics and Statistics, Central South Univer sity, Changsha 410083, Hunan, People’s Republic of China Email address : yuhe001@foxmail.com; yuhelingyun@foxmail.com
work page 2007
Reviewed August 14, 2026 · model on record in the stance chip above.
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