New Gosper-type Lambert series identities of levels 12 and 16

Russelle Guadalupe

arxiv: 2503.17795 · v3 · submitted 2025-03-22 · 🧮 math.NT

New Gosper-type Lambert series identities of levels 12 and 16

Russelle Guadalupe This is my paper

Pith reviewed 2026-05-22 23:07 UTC · model grok-4.3

classification 🧮 math.NT

keywords Lambert series identitiesGosper-type identitiesgeneralized eta-quotientscongruence subgroupsGamma_0(12)Gamma_0(16)q-seriesmodular forms

0 comments

The pith

New Gosper-type Lambert series identities of levels 12 and 16 equal sums of generalized eta-quotients on Gamma_0(12) and Gamma_0(16).

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

The paper derives new identities expressing Lambert series at levels 12 and 16 as q-series from eta-quotients. It constructs specific linear combinations of generalized eta-quotients on the groups Gamma_0(12) and Gamma_0(16) whose expansions match the desired forms. These extend earlier Gosper-type results known at smaller levels. A sympathetic reader cares because the identities tie infinite series directly to modular form properties on low-genus subgroups.

Core claim

We derive new Gosper-type Lambert series identities of levels 12 and 16 using certain sums of generalized η-quotients on the genus zero congruence subgroups Γ₀(12) and Γ₀(16).

What carries the argument

Linear combinations of generalized eta-quotients on the congruence subgroups Γ₀(12) and Γ₀(16) whose q-expansions take Lambert series form.

Load-bearing premise

The chosen linear combinations of generalized eta-quotients actually equal the claimed Lambert series forms rather than some other modular form whose q-expansion is not of Lambert type.

What would settle it

Expand the eta-quotient sum as a power series in q to order 200 and check whether every coefficient matches the corresponding term in the proposed Lambert series.

read the original abstract

We derive new Gosper-type Lambert series identities of levels $12$ and $16$ using certain sums of generalized $\eta$-quotients on the genus zero congruence subgroups $\Gamma_0(12)$ and $\Gamma_0(16)$.

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

Extends eta-quotient method to levels 12 and 16 for two new identities, but needs explicit coefficient match or uniqueness proof to confirm the Lambert series claim.

read the letter

The main point is that the paper produces two new Gosper-type Lambert series identities at levels 12 and 16 by taking linear combinations of generalized eta-quotients on Gamma0(12) and Gamma0(16). It follows the same pattern already used at other levels, so the work is an incremental extension rather than a new technique. If the calculations hold, it adds concrete examples that specialists might use when building tables or looking for patterns in q-series. The abstract states the results cleanly and the choice of groups makes sense given that they are genus zero. The soft spot is exactly the one flagged in the stress-test note. Showing that the combination is a modular form of the correct weight and level is not automatically enough when the space of forms has dimension greater than one. Without a coefficient-by-coefficient comparison or an independent argument that pins down the form uniquely, the equality to the claimed Lambert series could be off. The abstract supplies no derivation steps or sample expansions, so the reader has to trust that the full text contains those checks. This paper is aimed at people already working on eta-quotients and Lambert series identities. A reader in that narrow corner of number theory would find the new examples worth seeing, provided the verifications are present. It is coherent enough on its own terms to deserve a serious referee who can check the q-expansions and the matching argument directly.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to derive new Gosper-type Lambert series identities of levels 12 and 16 by expressing them as certain linear combinations of generalized η-quotients on the genus-zero groups Γ₀(12) and Γ₀(16).

Significance. If the claimed equalities hold, the identities would constitute a modest extension of the known list of Gosper-type series at small levels, consistent with existing techniques that exploit the low-dimensional spaces of modular forms on genus-zero congruence subgroups. No machine-checked proofs, reproducible code, or parameter-free derivations are mentioned.

major comments (1)

[Main theorem statements (presumably §3 or §4)] The central identification—that the indicated linear combination of generalized η-quotients equals a Lambert series of the stated form—requires either an explicit coefficient-by-coefficient q-expansion match up to a sufficient order or a uniqueness argument showing that the space of forms of the relevant weight and level on Γ₀(12) (resp. Γ₀(16)) is one-dimensional and spanned by the claimed Lambert series. Merely verifying modularity and weight is insufficient when dim > 1.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive comment on the verification of the main identities.

read point-by-point responses

Referee: [Main theorem statements (presumably §3 or §4)] The central identification—that the indicated linear combination of generalized η-quotients equals a Lambert series of the stated form—requires either an explicit coefficient-by-coefficient q-expansion match up to a sufficient order or a uniqueness argument showing that the space of forms of the relevant weight and level on Γ₀(12) (resp. Γ₀(16)) is one-dimensional and spanned by the claimed Lambert series. Merely verifying modularity and weight is insufficient when dim > 1.

Authors: We agree that establishing modularity and weight alone does not suffice for identification when the dimension of the space of forms exceeds one. The manuscript constructs the linear combination of generalized η-quotients to match the Lambert series and verifies that both sides transform correctly under the group action with the same weight. To strengthen the argument, the revised version will include an explicit coefficient-by-coefficient comparison of the q-expansions of both sides to a high order (sufficient to exceed the Sturm bound for the relevant space). We will also record the dimension of the space of forms of the given weight and level on Γ₀(12) and Γ₀(16) using the standard dimension formula, confirming it is one-dimensional in these cases. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on explicit modular-form identities without self-referential fitting or load-bearing self-citation

full rationale

The abstract and available description state that new Gosper-type Lambert series identities are derived from sums of generalized eta-quotients on Γ₀(12) and Γ₀(16). No equations, fitted parameters, or predictions appear in the provided text. No self-citations are invoked to justify uniqueness or to rename known results. The central step (equating eta-quotient sums to Lambert series) is presented as a derivation rather than a tautology or fit; without quoted equations reducing one quantity to another by construction, the chain is self-contained against external modular-form verification. This is the expected non-finding for a paper whose abstract contains no quantitative claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard facts about the Dedekind eta function, its transformation properties under congruence subgroups, and the definition of Lambert series; no free parameters, new entities, or ad-hoc axioms are visible.

axioms (2)

standard math The Dedekind eta function and its quotients transform as modular forms on Γ₀(N) for N=12 and N=16.
Required to define the sums that are asserted to equal the Lambert series.
domain assumption Γ₀(12) and Γ₀(16) are genus-zero groups, allowing eta-quotients to be expressed via hauptmoduls.
Stated directly in the abstract as the setting for the sums.

pith-pipeline@v0.9.0 · 5546 in / 1340 out tokens · 51457 ms · 2026-05-22T23:07:00.440443+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Z. S. Aygin, Introduction to Applications of Modular Forms, Computational Aspects , Synthesis Lectures on Mathematics & Statistics, Springer, Cham, 2023

work page 2023
[2]

El Bachraoui, On series identities of Gosper and integrals of Ramanujan theta function ψ(q), Proc

M. El Bachraoui, On series identities of Gosper and integrals of Ramanujan theta function ψ(q), Proc. Amer. Math. Soc. 147 (2019), 4451–4464

work page 2019
[3]

W. N. Bailey, Series of hypergeometric type which are infinite in both directions , Q. J. Math. os-7 (1936), 105–115

work page 1936
[4]

, A further note on two of Ramanujan’s formulae , Q. J. Math. 3 (1952), 158–160

work page 1952
[5]

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

work page 1991
[6]

B. Cho, J. K. Koo, and Y. K. Park, Arithmetic of the Ramanujan-G¨ ollinitz-Gordon continued fraction, J. Number Theory 129 (2009), 922–947

work page 2009
[7]

Cohen and F

H. Cohen and F. Str¨ omberg,Modular Forms, A Classical Approach , Graduate Studies in Mathematics, vol. 179, American Mathematical Society, Rhode Island, 2017

work page 2017
[8]

R. W. Gosper, Experiments and discoveries in q-trigonometry, Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (New York) (F. G. Garvan and M. E. H. Ismail, eds.), Springer, 2001, pp. 79–103

work page 2001
[9]

Guadalupe, Gosper’s lambert series of level 14, 2024, preprint at https://arxiv.org/abs/2412.18228

R. Guadalupe, Gosper’s lambert series of level 14, 2024, preprint at https://arxiv.org/abs/2412.18228

work page internal anchor Pith review arXiv 2024
[10]

He, Proofs for certain Πq identities of Gosper , J

B. He, Proofs for certain Πq identities of Gosper , J. Math. Anal. Appl. 492 (2020), 124486

work page 2020
[11]

, Proofs for a Πq-identity of Gosper , Adv. Appl. Math. 123 (2021), 102120

work page 2021
[12]

Paule and C.-S

P. Paule and C.-S. Radu, Holonomic relations for modular functions and forms: first guess, then prove , Int. J. Number Theory 17 (2021), 713–759

work page 2021
[13]

Radu, An algorithmic approach to Ramanujan-Kolberg identities , J

C.-S. Radu, An algorithmic approach to Ramanujan-Kolberg identities , J. Symbolic Comput. 68 (2015), 225–253

work page 2015
[14]

Wang, Modular proofs of Gosper’s identities , Adv

L. Wang, Modular proofs of Gosper’s identities , Adv. Appl. Math. 135 (2022), 102312

work page 2022
[15]

Yang, Transformation formulas for generalized Dedekind eta functions, Bull

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), 671–682

work page 2004
[16]

M. V. Yathirajsharma, On certain q-trigonometric identities analogous to that of Gosper’s , Rocky Mountain J. Math. 53 (2023), 629–646

work page 2023
[17]

M. V. Yathirajsharma, K. N. Harshitha, and K. R. Vasuki, On Gosper’s Πq and Lambert series identities , Hiroshima Math. J. 52 (2022), 113–137. Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines Email address: rguadalupe@math.upd.edu.ph

work page 2022

[1] [1]

Z. S. Aygin, Introduction to Applications of Modular Forms, Computational Aspects , Synthesis Lectures on Mathematics & Statistics, Springer, Cham, 2023

work page 2023

[2] [2]

El Bachraoui, On series identities of Gosper and integrals of Ramanujan theta function ψ(q), Proc

M. El Bachraoui, On series identities of Gosper and integrals of Ramanujan theta function ψ(q), Proc. Amer. Math. Soc. 147 (2019), 4451–4464

work page 2019

[3] [3]

W. N. Bailey, Series of hypergeometric type which are infinite in both directions , Q. J. Math. os-7 (1936), 105–115

work page 1936

[4] [4]

, A further note on two of Ramanujan’s formulae , Q. J. Math. 3 (1952), 158–160

work page 1952

[5] [5]

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

work page 1991

[6] [6]

B. Cho, J. K. Koo, and Y. K. Park, Arithmetic of the Ramanujan-G¨ ollinitz-Gordon continued fraction, J. Number Theory 129 (2009), 922–947

work page 2009

[7] [7]

Cohen and F

H. Cohen and F. Str¨ omberg,Modular Forms, A Classical Approach , Graduate Studies in Mathematics, vol. 179, American Mathematical Society, Rhode Island, 2017

work page 2017

[8] [8]

R. W. Gosper, Experiments and discoveries in q-trigonometry, Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (New York) (F. G. Garvan and M. E. H. Ismail, eds.), Springer, 2001, pp. 79–103

work page 2001

[9] [9]

Guadalupe, Gosper’s lambert series of level 14, 2024, preprint at https://arxiv.org/abs/2412.18228

R. Guadalupe, Gosper’s lambert series of level 14, 2024, preprint at https://arxiv.org/abs/2412.18228

work page internal anchor Pith review arXiv 2024

[10] [10]

He, Proofs for certain Πq identities of Gosper , J

B. He, Proofs for certain Πq identities of Gosper , J. Math. Anal. Appl. 492 (2020), 124486

work page 2020

[11] [11]

, Proofs for a Πq-identity of Gosper , Adv. Appl. Math. 123 (2021), 102120

work page 2021

[12] [12]

Paule and C.-S

P. Paule and C.-S. Radu, Holonomic relations for modular functions and forms: first guess, then prove , Int. J. Number Theory 17 (2021), 713–759

work page 2021

[13] [13]

Radu, An algorithmic approach to Ramanujan-Kolberg identities , J

C.-S. Radu, An algorithmic approach to Ramanujan-Kolberg identities , J. Symbolic Comput. 68 (2015), 225–253

work page 2015

[14] [14]

Wang, Modular proofs of Gosper’s identities , Adv

L. Wang, Modular proofs of Gosper’s identities , Adv. Appl. Math. 135 (2022), 102312

work page 2022

[15] [15]

Yang, Transformation formulas for generalized Dedekind eta functions, Bull

Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), 671–682

work page 2004

[16] [16]

M. V. Yathirajsharma, On certain q-trigonometric identities analogous to that of Gosper’s , Rocky Mountain J. Math. 53 (2023), 629–646

work page 2023

[17] [17]

M. V. Yathirajsharma, K. N. Harshitha, and K. R. Vasuki, On Gosper’s Πq and Lambert series identities , Hiroshima Math. J. 52 (2022), 113–137. Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines Email address: rguadalupe@math.upd.edu.ph

work page 2022