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arxiv: 2606.18110 · v1 · pith:WA5RZC7Qnew · submitted 2026-06-16 · 🧮 math.NT

Recursions for Mock Theta Functions

Matthew Ortiz , Martin Raum , Olav K. Richter This is my paper

Pith reviewed 2026-06-26 22:29 UTC · model grok-4.3

classification 🧮 math.NT
keywords mock theta functionsrecursionscoefficientsRankin-Cohen bracketsholomorphic projectionvector-valued modular formscusp forms
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The pith

The coefficients of Ramanujan's third order mock theta functions f and ω satisfy weighted recursions.

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

The paper establishes weighted recursions for the coefficients of Ramanujan's third order mock theta functions f and ω. It does so by applying a holomorphic projection operator to vector-valued Rankin-Cohen brackets of the completed mock theta series and their shadows. The vector-valued setting is used to exploit the vanishing of certain spaces of vector-valued cusp forms. If the claim holds, these recursions supply explicit relations that determine the coefficients in a recursive manner.

Core claim

We establish weighted recursions for the coefficients of Ramanujan's third order mock theta functions f and ω. Specifically, we apply a holomorphic projection operator to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows. By employing a vector-valued framework, we exploit the vanishing of certain spaces of vector-valued cusp forms. Our proof is AI-assisted and prioritizes accessibility.

What carries the argument

Holomorphic projection operator applied to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows.

If this is right

  • The coefficients of f and ω obey explicit weighted recursive relations.
  • The relations follow directly from the holomorphic projection of the indicated brackets.
  • The vanishing of the relevant cusp form spaces completes the proof of the recursions.
  • The method yields recursions specific to the third-order mock theta functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same projection technique might produce recursions for mock theta functions of other orders.
  • The recursions could be used to derive further q-series identities or congruences.
  • Independent numerical checks of the recursions for small n would provide a direct test.

Load-bearing premise

Certain spaces of vector-valued cusp forms vanish in the chosen vector-valued framework.

What would settle it

Direct computation of the coefficient of some power of q in the q-expansion of f or ω that violates the claimed weighted recursion.

read the original abstract

We establish weighted recursions for the coefficients of Ramanujan's third order mock theta functions $f$ and $\omega$. Specifically, we apply a holomorphic projection operator to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows. By employing a vector-valued framework, we exploit the vanishing of certain spaces of vector-valued cusp forms. Our proof is AI-assisted and prioritizes accessibility, allowing for straightforward customization and replication within the broader research community.

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

0 major / 2 minor

Summary. The manuscript claims to establish weighted recursions for the Fourier coefficients of Ramanujan's third-order mock theta functions f and ω. The method applies a holomorphic projection operator to vector-valued Rankin-Cohen brackets formed from completed mock theta series and their shadows; the vector-valued setting is used to invoke the vanishing of certain spaces of vector-valued cusp forms, yielding the recursions. The proof is described as AI-assisted and designed for accessibility and replication.

Significance. If the claimed recursions are correctly derived, the work would supply explicit, weighted relations among coefficients of well-studied mock theta functions, potentially useful for computation and for relating mock modular forms to classical modular forms. The vector-valued Rankin-Cohen bracket plus holomorphic projection technique is a recognized tool in the literature on mock modular forms; a successful application here would illustrate its utility for obtaining coefficient recursions without introducing free parameters.

minor comments (2)
  1. [Abstract] Abstract: the description of the proof strategy is given at a high level without any displayed equation, sample recursion, or explicit statement of the vanishing result invoked; adding one concrete instance of a derived recursion (even in an introductory paragraph) would make the central claim easier to verify and would align with the stated goal of accessibility.
  2. [Introduction] The manuscript states that the proof is AI-assisted; a brief clarification in the introduction or §1 on which steps were assisted (e.g., symbolic computation of brackets or verification of vanishing) would help readers assess reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on weighted recursions for the coefficients of Ramanujan's mock theta functions f and ω, and for the recommendation of minor revision. The report raises no specific major comments, so we have no points requiring detailed rebuttal or revision at this stage. We remain ready to incorporate any minor editorial suggestions identified by the editor.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives weighted recursions by applying the holomorphic projection operator to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows, then invoking the vanishing of certain vector-valued cusp form spaces. These are standard operators and vanishing statements drawn from the literature rather than self-referential definitions or fitted inputs. No equations or steps reduce by construction to the target recursions; the argument remains self-contained against external modular-form theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the holomorphic projection operator being well-defined on the vector-valued brackets and on the vanishing of the indicated cusp form spaces; both are standard domain facts in the theory of vector-valued modular forms rather than new postulates.

axioms (1)
  • domain assumption Vanishing of certain spaces of vector-valued cusp forms
    Invoked to conclude that the projection yields the desired recursions; location: abstract description of the vector-valued framework.

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discussion (0)

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Reference graph

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