Ninth degree analogue of Ramanujan's septic theta function identity

\"Ors Reb\'ak; Sun Kim

REVIEW 3 major objections 3 minor 25 references

Ninth degree analogue of Ramanujan's septic theta function identity

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves a ninth-degree analogue of the seventh-degree theta-function identity: the quotient φ(q^{1/9})/φ(q^9) splits as 1+u1+u2+u3+u4, with u1,u2,u4 coming from the roots of an explicit cubic.

desk verdict A genuine but incremental ninth-degree analogue; the root-ordering gap in Theorem 2.1 is real, trivially fixable, and the paper deserves refereeing. read the letter →

arxiv 2506.01181 v1 pith:T4O3G7VF submitted 2025-06-01 math.NT math.CA

Sun Kim , Örs Rebák This is my paper

classification math.NTmath.CA MSC 33C0505A3011F3211R29

keywords thetafunctionsninthdegreeidentitiesnoniclostnotebookclassinvariantsexplicitevaluationstrigonometricsepticfunctionidentity

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

The paper proves a ninth-degree analogue of the classic seventh-degree $\theta$-function identity from the lost notebook. The main theorem expresses the ratio φ($q^{{1/9}}$)/φ($q^{9}$) as 1+u1+u2+u3+u4, where u3 is the simpler ratio φ(q)/φ($q^{9}$)-1, the product u1u2u4 is an explicit infinite product, and u1,u2,u4 are cube roots of ratios of the roots of a cubic whose coefficients are built from u3 and that product. The proof adapts the dissection method used in the septic case, extracting pieces of the q-series according to the fractional part of the exponent. The paper also draws on an entry about cubic equations to reformulate the identity, and it gives five special evaluations of φ($e^{{-π√n}}$) in trigonometric form. These examples reproduce and unify several known values, add one new evaluation, and yield trigonometric identities parallel to ones in the original notebook.

What carries the argument

The machinery has two parts. The dissection identity of Lemma 2.2, applied to (a,b)=($q^{{1/9}}$,$q^{{1/9}}$), decomposes φ($q^{{1/9}}$) into nine summands; the symmetry of f(a,b) reduces them to the four u_k, giving Lemma 2.3. The second part is the operator M_α that selects the terms of a q-series whose exponents have a given fractional part; applying it to the cube of u1+u2+u4 produces the identities of Lemma 2.8, which show that α=$u2u4^{2}$, β=$u4u1^{2}$, γ=$u1u2^{2}$ are the elementary symmetric functions of the roots of r. Lemma 4.8 then fixes the root order for real q.

What would settle it

Take q=1/2. Compute the three roots of r(ξ) to high precision, order them by the two inequalities of Lemma 4.8, form u1,u2,u4 from the cube-root formulas, and check whether u1+u2+u4 equals φ($q^{{1/9}}$)/φ($q^{9}$)-1-u3 to the working precision. A mismatch at any q in (0,1) would show the ordering criterion is incomplete; agreement across a dense set of q would support the claimed identity.

Watch

Extended reading notes

Core claim

Theorem 2.1 is the central discovery. For 0<|q|<1, define u_k(q)=$2q^{{k^2/9}}$ f($q^{{9+2k}}$,$q^{{9-2k}}$)/φ($q^{9}$) for k=1,...,4. Then φ($q^{{1/9}}$)/φ($q^{9}$)=1+u1+u2+u3+u4, with u3=φ(q)/φ($q^{9}$)-1 and p:=u1u2u4=$8q^{{7/3}}$χ(q)/($χ^{6}$($q^{9}$)χ($q^{3}$)). If α,β,γ are the roots of r(ξ)=$ξ^{3}$-u3($φ^{2}$(q)/$φ^{2}$($q^{9}$)+3)($ξ^{2}$-$2u3^{2}$ξ)-$p^{3}$, then, with an appropriate ordering, u1=(βp/α)^{1/3}, u2=(γp/β)^{1/3}, u4=(αp/γ)^{1/3}. Lemma 4.8 supplies the needed ordering for real q in (0,1), and Theorems 4.9–4.13 use the identity to evaluate φ at five exponential arguments; Theorem 4.13 is presented as new.

Load-bearing premise

The result rests on being able to order the three roots of the cubic so that the cube-root formulas reproduce u1,u2,u4; the paper gives an ordering criterion only for real q between 0 and 1, and even there part of the criterion is checked by computer rather than by a closed-form proof.

Editorial extensions

If this is right

For real q in (0,1), the identity reduces the computation of φ(q^{1/9})/φ(q^9) to one infinite product p, the ratio φ(q)/φ(q^9), and the ordered roots of a single cubic.
The examples in Theorems 4.9–4.13 give five explicit values of φ(e^{-π√n}); the first four reproduce, in trigonometric form, values already known in radical form, while Theorem 4.13 is new.
Comparing the trigonometric and radical forms yields several families of trigonometric identities, including analogues of the identities cos(2π/9)+cos(4π/9)=cos(π/9) and 1/cos(4π/9)+1/cos(2π/9)=1/cos(π/9)+6.
Theorem 3.3 gives an alternative formulation using the cubic-equation entry: φ(q^{1/9})/φ(q^9)=1+u3+(py)^{1/3}, where y satisfies a quadratic whose coefficients are rational in u3 and p.
The paper's closing suggestion is that these results fit into a larger family of theta-function identities of higher degree envisioned in Section 12 of Chapter 20 of the second notebook.

Reading between the lines

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

The root-ordering gap for complex q could probably be closed by fixing cube-root branches via analytic continuation from the real interval; if so, Theorem 2.1 would become a genuine q-series identity on the whole punctured unit disk, not merely a statement for real q.
The same fractional-part extraction method should apply to higher odd degrees: for degree 11 one would expect five u_k terms and a quartic whose roots play the role of α,β,γ; the pattern α=u2u4^2 hints at a general prescription tied to the divisors of the degree.
The value of G_729 is known from the literature, and the paper notes that the corresponding cubic is much more complicated; a concrete next step is to find a transformation that makes the root ordering tractable for n=1/729.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is a legitimate ninth-degree analogue of Ramanujan's septic identity, not a repackaging of the septic case. The main identity in Theorem 2.1 is new, and the derivation is mostly self-contained: the dissection in Lemma 2.3, the operator argument in Lemma 2.8, and the modular equation inputs are all explicit. The five evaluations include one genuinely new value (Theorem 4.13); the first four duplicate known results, and the authors say so. I'd rate this as a useful contribution for q-series and modular equation people, though not a field-changer.

The real issue is the one the stress-test flags: Theorem 2.1(iv) does not specify which root of r is labelled alpha, beta, gamma, and for complex q the equalities as written aren't determinate. But it's a small fix. Immediately after the theorem the authors state (6), alpha = u2 u4^2, etc., which is a valid ordering of the roots for all 0<|q|<1. The proof of (iv) actually assumes (6). So the theorem is not false; it's just incomplete as stated. A referee should ask that (6) be incorporated into the theorem statement, or that the theorem say "for the ordering defined by (6)". That resolves the issue entirely.

Second soft spot: several example proofs (Theorems 4.10–4.13) rely on "Mathematica confirms" for the root-order conditions and for one polynomial identity. That's a bit casual. These are finite verifications and likely correct, but a short derivation or a computer-readable file would be better. It doesn't undermine the central proof, which doesn't depend on those examples.

Citation pattern is healthy. The paper builds on Son's septic proof and the second author's earlier work; those citations are earned. The acknowledgement of prior evaluations is honest.

Net: the core claim holds up. With the ordering made explicit and the Mathematica checks documented, this is a publishable paper. I'd send it to a refereed journal; the requested revisions are minor.

Recommendation: yes, engage. It's a solid, incremental addition.

Referee Report

3 major / 3 minor

Summary. The paper proposes a ninth-degree analogue of Ramanujan's septic theta-function identity. Theorem 2.1 states that for |q|<1 the quotient φ(q^{1/9})/φ(q^9) equals 1+u_1+u_2+u_3+u_4, with explicit product and sum formulas p=u_1u_2u_4 and u_3=φ(q)/φ(q^9)-1, and expresses u_1,u_2,u_4 as cube roots of ratios of the roots α,β,γ of a cubic r(ξ). Section 3 applies Ramanujan's cubic-entry identity from the second notebook to restate the main result, and Section 4 gives five explicit evaluations of φ(e^{-π√n}) in trigonometric form, including one new example. The proofs are based on standard q-series identities, the fractional-exponent operator M_α, class invariants, and exact trigonometric reductions.

Significance. The paper gives a substantial, explicit extension of a classical Ramanujan identity, and the derivation of parts (i)-(iii) of Theorem 2.1 is clean and checkable. The connection with Ramanujan's cubic-entry identity is elegant, and the five explicit quotient evaluations, one of which is new, are useful concrete data. The main result is not circular: p and u_3 are functions of q, and the underlying algebra is sound. The principal weaknesses are a determinacy gap in the statement of Theorem 2.1(iv), a factor-of-three typo in Theorem 4.9, and reliance on undocumented Mathematica checks in several example proofs. These are fixable within the paper's scope.

major comments (3)

[Theorem 2.1, parts (iv)-(v) and Eq. (6)] Part (iv) is not determinate as stated for 0<|q|<1. The roots α,β,γ of r are unordered, and the displayed formulas for u_1,u_2,u_4 change under permutation of the roots. The proof invokes Eq. (6) to fix the ordering, but Eq. (6) is not part of Theorem 2.1 and no cube-root branch convention is specified (each ratio such as βp/α equals u_1^3, so the equality holds only up to a cube root of unity). Lemma 4.8 supplies an ordering criterion only for real 0<q<1. Please incorporate Eq. (6) and an explicit cube-root branch into Theorem 2.1, or restrict the statement to the real-q setting in which Lemma 4.8 applies.
[Theorem 4.9 and Eq. (22)] The displayed formula in Theorem 4.9 is inconsistent with the proof by a factor of 3. Combining Eq. (22) with Theorem 2.1(i) gives φ(e^{-27π})/φ(e^{-3π}) = (1/(3√3))(1+u_1+u_2+u_3+u_4) = 1/3 + p^{1/3}S/(3√3), where S is the sum of the three cube-root terms and p^{1/3} = (16(11√3-19))^{1/9}. This is (1/3){1 + p^{1/3}S/√3}. The printed statement has √3 instead of 1/√3 inside the braces, which makes the stated value three times too large. Please correct the theorem statement.
[Theorems 4.10, 4.11, 4.12, and 4.13] The proofs of these four theorems rely on undocumented 'Mathematica confirms' statements: in Theorems 4.10 and 4.11 for the conditions of Lemma 4.8(i)-(ii), and in Theorems 4.12 and 4.13 additionally for the identification of the displayed α,β,γ as roots of r. Since these are exact identities, a formal proof should include either a complete exact derivation or a minimal reproducible computation. Please add the supporting verification.

minor comments (3)

[Notation after Theorem 2.1] The label 'Theorem 2.1(0)' for the definition of u_k is confusing; consider calling it 'part (0)' or moving the definition out of the numbered theorem.
[Lemma 4.8, sentence after the proof] The sentence 'Condition (ii) is needed to fulfill Theorem 2.1(0)' is misleading; condition (ii) is used to ensure the ordering u_1>u_2>u_4 stated in Lemma 4.6, not to fulfill the definition in Theorem 2.1(0).
[Final paragraph, page 20] There is a typo in 'septic theta function identitiy'; it should be 'identity'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Theorem 2.1 is derived from classical theta identities, with an explicit root construction rather than a fitted input.

full rationale

The central identity is not circular. Theorem 2.1(i) is proved from Entry 31 of Ramanujan's second notebook via Lemma 2.2 and classical theta-function identities; the quantities p and u3 are defined directly as q-series, not fitted to the examples. Part (iv) is established constructively by defining α := u2 u4^2, β := u4 u1^2, γ := u1 u2^2 in equation (6); the cube-root formulas then follow algebraically from p = u1 u2 u4, and the symmetric sums of α, β, γ are evaluated using Lemmas 2.8(i),(ii) and the product p^3 to match the cubic (v). This is a constructive algebraic identity, not a prediction derived from the theorem it is meant to prove. Lemma 2.8 is obtained by applying the series-extraction operator M_α to an identity derived from classical transformation formulas, not from Theorem 2.1. The examples in Section 4 use known class invariants and verify the root ordering via Lemma 4.8 and Mathematica checks; they are not fitted values. The only notable self-citation is [18, Lemma 3.2], used in Lemma 4.6 for monotonicity of the u_k in the ordering lemma; this concerns root selection for examples, not the derivation of the identity, and it is an independent published result. The under-specification of the root order in Theorem 2.1(iv) for non-real q is a correctness and clarity issue, not a circularity: the proof supplies equation (6) as an ordering, and Lemma 4.8 gives sufficient conditions for real q. Score 2 reflects the presence of minor self-citations in auxiliary lemmas, not circular dependence of the main claim.

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

The paper introduces no free parameters fitted to data and no invented physical entities. Its central claim rests on classical theta function and modular equation identities, plus local computer algebra checks in the examples.

assumptions (5)

standard math Jacobi triple product identity
Used in Lemma 2.5 and throughout to pass between series and product representations of theta functions.
domain assumption Ramanujan's modular equation Entry 1(iii) from the second notebook
Invoked in Lemma 2.4(ii) to relate phi^4(q^3)/phi^4(q^9) to u3, a load-bearing input for the later identities.
domain assumption Known class invariant values G_n from Ramanujan and Watson
Used in the examples to compute p and u3 at specific q values.
standard math Power series uniqueness for the extraction operator M_alpha
Used in Lemma 2.8 to derive identities by collecting terms with fixed fractional exponent.
domain assumption Entry 31 of Chapter 16 of Ramanujan's second notebook
Used in Lemma 2.2 and Lemma 2.3 to establish the nonic dissection of phi(q^(1/9)).

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Cite this review

Pith. "Pith review of Ninth degree analogue of Ramanujan's septic theta function identity." pith.science (2026). https://pith.science/paper/T4O3G7VF

@misc{pith2026250601181,
  author       = {Pith},
  title        = {Pith review of: Ninth degree analogue of Ramanujan's septic theta function identity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4O3G7VF}},
  note         = {Machine review of arXiv:2506.01181}
}

abstract

On page 206 in his lost notebook, Ramanujan recorded a seventh degree identity for his theta function $\varphi(q)$. We give an analogous ninth degree identity. We also provide an application of an entry from his second notebook on a cubic equation and an interpretation with theta functions for some of his trigonometric identities. Lastly, we calculate five examples for $\varphi(e^{-\pi\sqrt{n}})$.

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

G. E. Andrews and B. C. Berndt.Ramanujan’s Lost Notebook, Part II.Springer, New York, 2009

work page 2009
[2]

B. C. Berndt.Ramanujan’s Notebooks, Part III.Springer-Verlag, New York, 1991

work page 1991
[3]

B. C. Berndt.Ramanujan’s Notebooks, Part IV.Springer-Verlag, New York, 1994

work page 1994
[4]

B. C. Berndt.Ramanujan’s Notebooks, Part V.Springer-Verlag, New York, 1998

work page 1998
[5]

B. C. Berndt and S. Bhargava. Ramanujan—For Lowbrows.Amer. Math. Monthly, 100(7):644–656, 1993

work page 1993
[6]

B. C. Berndt and H. H. Chan. Ramanujan’s explicit values for the classical theta function.Mathematika, 42(2):278–294, 1995

work page 1995
[7]

B. C. Berndt, S. Kim, and A. Zaharescu. Exact evaluations and reciprocity theorems for finite trigonometric sums.Res. Math. Sci., 10(4):article number 40, 2023

work page 2023
[8]

B. C. Berndt and Ö. Rebák. Explicit values for Ramanujan’s theta functionφ(q).Hardy–Ramanujan J., 44(1):41–50, 2021

work page 2021

Show all 25 references

[9]

B. C. Berndt and Ö. Rebák. Cubic and quintic analogues of Ramanujan’s septic theta function identity. Ramanujan J., 66(4):article number 66, 2025

2025
[10]

J. M. Borwein and P. B. Borwein.Pi and the AGM. Wiley, New York, 1987

1987
[11]

J. M. Borwein and I. J. Zucker. Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind.IMA J. Numer. Anal., 12(4):519–526, 1992

1992
[12]

I. S. Gradshteyn and I. M. Ryzhik.Table of Integrals, Series, and Products. 7th ed., A. Jeffrey and D. Zwillinger, eds., Academic Press, New York, 2007

2007
[13]

C. G. J. Jacobi.Fundamenta nova theoriae functionum ellipticarum. Sumptibus fratrum Borntræger, Re- giomonti, 1829
[14]

Ramanujan

S. Ramanujan. Modular equations and approximations toπ.Quart. J. Math., 45:350–372, 1914

1914
[15]

Ramanujan.Collected Papers of Srinivasa Ramanujan

S. Ramanujan.Collected Papers of Srinivasa Ramanujan. G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, eds., Cambridge University Press, Cambridge, 1927

1927
[16]

Ramanujan.Notebooks of Srinivasa Ramanujan, Volume II.Tata Institute of Fundamental Research, Bombay, 1957

S. Ramanujan.Notebooks of Srinivasa Ramanujan, Volume II.Tata Institute of Fundamental Research, Bombay, 1957

1957
[17]

Ramanujan.The Lost Notebook and Other Unpublished Papers

S. Ramanujan.The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi, 1988

1988
[18]

Ö.Rebák.ThethreemissingtermsinRamanujan’ssepticthetafunctionidentity.Ramanujan J., 60(4):885– 911, 2023

2023
[19]

S. H. Son. Septic theta function identities in Ramanujan’s lost notebook.Acta Arith., 98(4):361–374, 2001

2001
[20]

S. H. Son. Circular summations of theta functions in Ramanujan’s lost notebook.Ramanujan J., 8(2):235– 272, 2004

2004
[21]

S. H. Son. Ramanujan’s symmetric theta functions in his lost notebook. InSpecial Functions and Orthogo- nal Polynomials, D. Dominici and R. S. Maier, eds., Contemp. Math., Vol. 471, pp. 187–202. American Mathematical Society, Providence, RI, 2008

2008
[22]

G. N. Watson. Singular moduli (3).Proc. London Math. Soc., 40(1):83–142, 1936

1936
[23]

E. T. Whittaker and G. N. Watson.A Course of Modern Analysis. 4th ed., Cambridge University Press, Cambridge, 1950

1950
[24]

Yi.The Construction and Applications of Modular Equations

J. Yi.The Construction and Applications of Modular Equations. PhD thesis, University of Illinois at Urbana–Champaign, Urbana, Illinois, 2001

2001
[25]

I. J. Zucker. The evaluation in terms ofΓ-functions of the periods of elliptic curves admitting complex multiplication.Math. Proc. Cambridge Philos. Soc., 82(1):111–118, 1977. Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk Na- tional Universi...

1977

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

G. E. Andrews and B. C. Berndt.Ramanujan’s Lost Notebook, Part II.Springer, New York, 2009

work page 2009

[2] [2]

B. C. Berndt.Ramanujan’s Notebooks, Part III.Springer-Verlag, New York, 1991

work page 1991

[3] [3]

B. C. Berndt.Ramanujan’s Notebooks, Part IV.Springer-Verlag, New York, 1994

work page 1994

[4] [4]

B. C. Berndt.Ramanujan’s Notebooks, Part V.Springer-Verlag, New York, 1998

work page 1998

[5] [5]

B. C. Berndt and S. Bhargava. Ramanujan—For Lowbrows.Amer. Math. Monthly, 100(7):644–656, 1993

work page 1993

[6] [6]

B. C. Berndt and H. H. Chan. Ramanujan’s explicit values for the classical theta function.Mathematika, 42(2):278–294, 1995

work page 1995

[7] [7]

B. C. Berndt, S. Kim, and A. Zaharescu. Exact evaluations and reciprocity theorems for finite trigonometric sums.Res. Math. Sci., 10(4):article number 40, 2023

work page 2023

[8] [8]

B. C. Berndt and Ö. Rebák. Explicit values for Ramanujan’s theta functionφ(q).Hardy–Ramanujan J., 44(1):41–50, 2021

work page 2021