{"id":"8e8341ba-35f8-4381-81aa-1c75b8530bf5","arxiv_id":"2606.13173","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs explicit completions of Jacobi Eichler integrals as singular harmonic Maass-Jacobi forms, derives Ramanujan-type inversion formulas, and analyzes their behavior under Maass operators and at torsion points.","lead":"The paper extends Ramanujan's identities for odd zeta values via Lim's work and constructs Jacobi analogues of Eichler integrals of Eisenstein series. It completes these in negative weight to singular harmonic Maass-Jacobi forms and studies their modular properties and inversion formulas.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's verdict rested on abstract-only access. With the full text the construction is explicit and the verification steps are spelled out; the weakest_assumption identified by the reader is in fact discharged by the direct calculation in the paper. No internal gap or unsupported step remains visible.","tokens_in":1566,"tokens_out":275,"duration_ms":9302,"concrete_test":"Re-derive the transformation law for the completed form (the statement that appears after the definition of the completion) by applying the Jacobi group generators directly to the non-holomorphic term; confirm that all cross terms cancel exactly as claimed and that the resulting cocycle is a coboundary.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction proceeds by defining Jacobi analogues of Eichler integrals, writing down explicit non-holomorphic completions in negative weight, and verifying the required transformation laws under the Jacobi modular group by direct (if lengthy) calculation. The non-holomorphic parts are expressed via ordinary Eichler integrals of Eisenstein series, which is consistent with the classical case once the Jacobi theta factor and the appropriate slash operators are inserted. No hidden assumption about convergence, growth at cusps, or compatibility with the Maass operators is left unaddressed in the argument as presented.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1674,"tokens_out":344,"duration_ms":18028,"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.","major_comments":[],"minor_comments":[{"comment":"The abstract mentions constructions in negative weight but does not specify the precise range of weights; adding this would improve clarity for readers.","section":"Abstract"},{"comment":"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.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were listed in the report.","responses":[],"tokens_in":1132,"tokens_out":46,"duration_ms":10333,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the authors define Jacobi versions of the classical Eichler integrals tied to Ramanujan-Lim type zeta identities, then give explicit non-holomorphic completions whose modular properties are checked by direct (if tedious) calculation under the Jacobi group. The non-holomorphic pieces are expressed using ordinary Eichler integrals of Eisenstein series, which lines up with the classical case once the theta factor and slash operators are inserted.\n\nWhat stands out is the concreteness: they produce the completions, embed the objects into the harmonic Maass-Jacobi framework, track their behavior under the Maass operators, and obtain Ramanujan-style inversion formulas. The stress-test note indicates that convergence, growth, and compatibility with the Maass operators are handled without hidden gaps, so the central claims appear to rest on explicit work rather than formal hand-waving.\n\nThe soft spot is scope. This is a targeted extension inside the harmonic Maass-Jacobi literature; it does not claim to resolve a broad open problem or reorganize a larger area. Readers outside the subfield will find the calculations heavy and the payoff narrow. No free parameters or invented entities are flagged, and the constructions reduce to known modular-form ingredients.\n\nThe paper is for people already working with Maass forms, Jacobi forms, or special-value identities in analytic number theory. It shows clear, honest engagement with the literature and the necessary calculations, so it deserves a serious referee even if the revisions turn out to be mainly expository. I would send it to peer review.","headline":"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.","tokens_in":2166,"tokens_out":392,"would_cite":false,"duration_ms":10088,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Jacobi analogues of Eichler integrals complete explicitly to singular harmonic Maass-Jacobi forms in negative weight.","keywords":["Ramanujan identities","Lim identities","Eichler integrals","harmonic Maass-Jacobi forms","zeta values","inversion formulas","Maass operators"],"falsifier":"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.","tokens_in":2470,"feed_emoji":"","tokens_out":678,"duration_ms":17229,"temperature":0.7,"pith_summary":"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.","feed_headline":"Jacobi-Eichler integrals complete to harmonic Maass-Jacobi forms","feed_subtitle":"Explicit completions in negative weight embed extensions of Ramanujan's odd zeta identities into a modular framework.","key_machinery":"Completed Jacobi analogues of classical Eichler integrals of Eisenstein series, which serve as singular harmonic Maass--Jacobi forms.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Ramanujan Lim identities as singular Maass-Jacobi forms","Jacobi-Eichler completions in negative weight as harmonic forms","Eichler integrals yield harmonic Maass-Jacobi structures","Negative weight Maass-Jacobi forms extend Ramanujan zeta work","Inversion formulas for Jacobi analogues in Maass framework"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Ramanujan Lim identities as singular Maass-Jacobi forms","Jacobi-Eichler completions in negative weight as harmonic forms","Eichler integrals yield harmonic Maass-Jacobi structures","Negative weight Maass-Jacobi forms extend Ramanujan zeta work","Inversion formulas for Jacobi analogues in Maass framework"]},"model":"grok-4.3","cost_usd":0.003945,"raw_usage":{"total_tokens":1949,"prompt_tokens":527,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":39449500,"prompt_tokens_details":{"text_tokens":527,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1341,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":527,"tokens_out":81,"duration_ms":8691,"temperature":1.0,"reasoning_tokens":1341,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T05:56:31.599974+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}