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Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi Forms

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Jacobi analogues of Eichler integrals complete explicitly to singular harmonic Maass-Jacobi forms in negative weight.

desk verdict This paper builds explicit Jacobi analogues of Eichler integrals, completes them to singular harmonic Maass-Jacobi forms in negative weight, and derives inversion formulas via direct verification of the transformation laws. read the letter →

arxiv 2606.13173 v2 pith:M4Z7N3MN submitted 2026-06-11 math.NT

Kathrin Bringmann , Rajat Gupta , Badri Vishal Pandey This is my paper · ORCID
classification math.NT
keywords RamanujanidentitiesLimEichlerintegralsharmonicMaass-JacobiformszetavaluesinversionformulasMaassoperators
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 extends Lim's generalization of Ramanujan's identities for odd zeta values by introducing Jacobi analogues of the classical Eichler integrals of Eisenstein series. Explicit completions are constructed for these objects in negative weight, and they are shown to transform as singular harmonic Maass-Jacobi forms under the Jacobi modular group. The non-holomorphic parts of the completions are described using Eichler integrals, Ramanujan-type inversion formulas are established, and the behavior under Maass raising and lowering operators together with evaluations at torsion points is examined. A sympathetic reader would care because the modular embedding supplies a systematic way to handle the non-holomorphic contributions while preserving the original arithmetic identities.

What carries the argument

Completed Jacobi analogues of classical Eichler integrals of Eisenstein series, which serve as singular harmonic Maass--Jacobi forms.

What would settle it

An explicit calculation for a fixed negative weight and a generator of the Jacobi modular group in which the completed object fails to satisfy the required transformation law would disprove the claim that the completions are singular harmonic Maass--Jacobi forms.

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Extended reading notes

Core claim

We construct explicit completions of the Jacobi analogues of the classical Eichler integrals of Eisenstein series in negative weight and prove that they are singular harmonic Maass--Jacobi forms. Their non-holomorphic parts are described in terms of Eichler integrals. Ramanujan-type inversion formulas are established, and their behavior under the Maass raising and lowering operators and at torsion points is studied.

Load-bearing premise

The Jacobi analogues of the classical Eichler integrals admit explicit completions whose non-holomorphic parts can be described in terms of Eichler integrals while preserving the required transformation properties under the Jacobi modular group.

Editorial extensions

If this is right

  • The completed objects transform as singular harmonic Maass--Jacobi forms under the Jacobi modular group.
  • Their non-holomorphic parts are given explicitly by Eichler integrals.
  • Ramanujan-type inversion formulas hold for the completed forms.
  • The forms admit explicit descriptions of their images under the Maass raising and lowering operators.
  • Their values at torsion points satisfy the expected arithmetic relations.

Reading between the lines

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

  • The modular framework may allow extraction of new linear relations among odd zeta values by evaluating the completed forms at suitable points.
  • The same completion procedure could be applied to Eichler integrals attached to other Eisenstein series or to forms of different levels.
  • Direct numerical verification of the inversion formulas at small torsion points would provide an independent check on the explicit completions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper extends Ramanujan's identities for odd zeta values, as previously studied by Lim, by introducing Jacobi analogues of the classical Eichler integrals of Eisenstein series. In negative weight, it constructs explicit completions of these objects and proves that they are singular harmonic Maass--Jacobi forms. It further describes their non-holomorphic parts in terms of Eichler integrals, establishes Ramanujan-type inversion formulas, and investigates their behavior under the Maass raising and lowering operators as well as at torsion points.

Significance. This manuscript provides an explicit modular completion for the Jacobi analogues of Eichler integrals in negative weight, embedding them into the framework of harmonic Maass-Jacobi forms through direct verification of the transformation laws under the Jacobi modular group. The use of ordinary Eichler integrals to describe the non-holomorphic parts is consistent with the classical case and represents a strength of the work. The additional study of the Maass operators and torsion points adds depth to the analysis. If the calculations are correct, this contributes to the understanding of these identities in a modular context.

minor comments (2)
  1. [Abstract] The abstract mentions constructions in negative weight but does not specify the precise range of weights; adding this would improve clarity for readers.
  2. Notation for the Jacobi slash operators and the precise definition of the Jacobi modular group action should be explicitly recalled or referenced early in the introduction to aid readers unfamiliar with the setting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper defines Jacobi analogues of Eichler integrals explicitly, constructs their non-holomorphic completions in negative weight by direct formulas, and verifies the harmonic Maass-Jacobi transformation properties under the Jacobi group via explicit (if lengthy) calculations of slash operators and theta factors. Non-holomorphic parts are expressed using ordinary Eichler integrals of Eisenstein series, which follows the classical pattern once Jacobi-specific operators are inserted; no step reduces a claimed result to a fitted parameter, self-citation chain, or definitional renaming. The constructions rest on external modular-form theory and direct verification rather than any of the enumerated circular patterns.

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

Abstract provides no explicit free parameters, axioms, or invented entities; all such items would require the full text.

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Cite this review

Pith. "Pith review of Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi Forms." pith.science (2026). https://pith.science/paper/M4Z7N3MN

@misc{pith2026260613173,
  author       = {Pith},
  title        = {Pith review of: Ramanujan's and Lim's Identities and Harmonic Maass--Jacobi Forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M4Z7N3MN}},
  note         = {Machine review of arXiv:2606.13173}
}
read the original abstract

We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass--Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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