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arxiv: 2605.15740 · v1 · pith:JBXXBRNEnew · submitted 2026-05-15 · 🧮 math.NT · math.GT· math.QA

Bailey pairs, Eichler integrals and unified Witten-Reshetikhin-Turaev invariants

Jeremy Lovejoy , Robert Osburn , Matthias Storzer This is my paper

Pith reviewed 2026-05-20 16:22 UTC · model grok-4.3

classification 🧮 math.NT math.GTmath.QA
keywords Bailey pairsEichler integralsWitten-Reshetikhin-Turaev invariantsq-multisumsroots of unitymodular formshomology spheresquantum invariants
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The pith

q-multisums at roots of unity equal limiting values of Eichler integrals of weight 3/2 modular forms

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

The paper constructs infinite families of identities equating q-multisums evaluated at roots of unity to the limiting values of Eichler integrals of weight 3/2 modular forms. These identities recover all of Hikami's earlier results and conjectures on q-series tied to unified Witten-Reshetikhin-Turaev invariants and extend the Lawrence-Zagier expression for the Poincaré homology sphere. A reader would care because the work supplies a uniform method to connect q-hypergeometric series directly to analytic continuations of modular forms, which may streamline calculations involving quantum invariants of three-manifolds.

Core claim

Using the Bailey pair machinery together with a new relation between incomplete quadratic Gauss sums that carry periodic coefficients, the authors prove that for infinite families of q-multisums the value at a root of unity coincides with the limit of the Eichler integral of a weight 3/2 modular form. The construction recovers every result and conjecture of Hikami and generalizes the Lawrence-Zagier theorem that expresses the WRT invariant of the Poincaré homology sphere in this manner.

What carries the argument

Bailey pair machinery combined with a novel identity for incomplete quadratic Gauss sums with periodic coefficients, which converts the q-multisums into series whose limits match the Eichler integrals

Load-bearing premise

The new relation for incomplete quadratic Gauss sums with periodic coefficients holds and combines with Bailey pairs to yield the claimed identities for the selected families of q-multisums.

What would settle it

A concrete q-multisum family together with a root of unity at which the multisum value fails to equal the corresponding Eichler integral limit, or an explicit counterexample to the periodic-coefficient Gauss-sum relation.

read the original abstract

In 1999, Lawrence and Zagier expressed the Witten-Reshetikhin-Turaev (WRT) invariant of the Poincar\'e homology sphere as the limiting value of the Eichler integral of a weight 3/2 modular form. Habiro's construction of the unified WRT invariant subsequently recast this result as an identity for a $q$-hypergeometric series at roots of unity. This motivated Hikami to prove analogous $q$-series identities involving the unified WRT invariants of certain Brieskorn homology spheres. Hikami also made several conjectures of a similar type for $q$-series with no apparent connection to quantum invariants. In this paper we use the Bailey pair machinery and a novel relation between incomplete quadratic Gauss sums with periodic coefficients to construct infinite families of identities between $q$-multisums at roots of unity and limiting values of Eichler integrals of weight 3/2 modular forms. These identities include all of Hikami's results and conjectures as well as a generalization of the result of Lawrence and Zagier.

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.

Referee Report

1 major / 1 minor

Summary. The paper claims to combine the Bailey pair transformation machinery with a novel identity relating incomplete quadratic Gauss sums carrying periodic coefficients, thereby producing infinite families of evaluations of q-multisums at roots of unity that equal the limiting values of Eichler integrals of weight-3/2 modular forms. The resulting identities are asserted to recover every result and conjecture of Hikami together with a generalization of the Lawrence–Zagier evaluation for the Poincaré homology sphere.

Significance. If the construction is valid, the work supplies a uniform generating mechanism for a class of q-series identities that had previously appeared piecemeal, thereby linking the theory of unified WRT invariants to the analytic properties of weight-3/2 Eichler integrals in a systematic way. The explicit use of Bailey pairs is a methodological strength that may allow further extensions.

major comments (1)
  1. [Statement and derivation of the novel Gauss-sum relation] The central construction rests on a newly stated relation for incomplete quadratic Gauss sums with periodic coefficients. Because this relation is required to hold for arbitrary periods and arbitrary families of q-multisums in order to deliver the claimed infinite families (including all of Hikami’s conjectures), the manuscript must supply a complete, self-contained derivation of the relation together with an explicit statement of its range of validity. Any hidden restriction on the period or on the multisum parameters would prevent the Bailey-pair step from producing the asserted identities in full generality.
minor comments (1)
  1. Notation for the periodic coefficients in the Gauss sums and for the parameters of the q-multisums should be made uniform across the statements of the main theorems and the examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive suggestion concerning the presentation of the novel Gauss-sum relation. We address this point below and will revise the paper accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Statement and derivation of the novel Gauss-sum relation] The central construction rests on a newly stated relation for incomplete quadratic Gauss sums with periodic coefficients. Because this relation is required to hold for arbitrary periods and arbitrary families of q-multisums in order to deliver the claimed infinite families (including all of Hikami’s conjectures), the manuscript must supply a complete, self-contained derivation of the relation together with an explicit statement of its range of validity. Any hidden restriction on the period or on the multisum parameters would prevent the Bailey-pair step from producing the asserted identities in full generality.

    Authors: We agree that a fully self-contained derivation and an explicit statement of the range of validity are necessary to substantiate the generality of the construction. In the revised manuscript we will insert a dedicated subsection (or appendix) that derives the relation for incomplete quadratic Gauss sums carrying periodic coefficients from first principles, without relying on external results for the core steps. We will also state the precise hypotheses under which the identity holds, confirming that it applies for arbitrary periods and for the full range of multisum parameters appearing in the Bailey-pair iterations. This will explicitly rule out hidden restrictions and ensure that the subsequent transformations recover all of Hikami’s results and conjectures as asserted. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation combines established Bailey pairs with a novel Gauss-sum relation

full rationale

The paper derives the claimed identities by applying the Bailey pair machinery (an established technique from prior literature) together with a newly stated relation on incomplete quadratic Gauss sums carrying periodic coefficients. The abstract and structure present this Gauss-sum relation as an original contribution whose validity enables the construction of the infinite families, including generalizations of Hikami and Lawrence–Zagier. No step reduces the target q-multisum evaluations at roots of unity to a fitted parameter, a self-citation chain, or a redefinition of the desired result; the central identities are obtained as consequences rather than presupposed inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of Bailey pairs, Eichler integrals, and modular forms of weight 3/2, together with one newly introduced relation on incomplete quadratic Gauss sums. No free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard transformation and convergence properties of Bailey pairs and Eichler integrals of weight 3/2 modular forms hold.
    Invoked implicitly when applying the Bailey pair machinery to produce the q-series identities at roots of unity.

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Reference graph

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