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Functional equations and an Euler-product obstruction for mixed mock L-series

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and notation.
A mixed mock modular form combines mock-modular and modular factors under a specified convention; its nonholomorphic completion encodes the shadow. For a Fourier expansion $f(\tau)=\sum a_nq^n$, an L-series is a Dirichlet series built from its coefficients, typically $\sum_{n\ge1}a_n n^{-s}$ in a region of convergence. Growth, cusps and completion may affect analytic continuation and functional equations.
Catalogue wording retained for traceability.
Do the L-series of mixed mock modular forms have interesting properties?
Scope and source qualification.
The source asks about interesting properties without prescribing a single theorem. No Euler product, universal functional equation or convergence condition is silently imposed. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement. The statement covers the original source questions rather than only the title’s subcase.

Short English statement

Identify significant analytic or arithmetic properties of L-series attached to mixed mock modular forms.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-144-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-144

Functional equations and an Euler-product obstruction for mixed mock L-series

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and notation.
A mixed mock modular form combines mock-modular and modular factors under a specified convention; its nonholomorphic completion encodes the shadow. For a Fourier expansion $f(\tau)=\sum a_nq^n$, an L-series is a Dirichlet series built from its coefficients, typically $\sum_{n\ge1}a_n n^{-s}$ in a region of convergence. Growth, cusps and completion may affect analytic continuation and functional equations.
Catalogue wording retained for traceability.
Do the L-series of mixed mock modular forms have interesting properties?
Scope and source qualification.
The source asks about interesting properties without prescribing a single theorem. No Euler product, universal functional equation or convergence condition is silently imposed. The catalogue title/status was paired with a machine-generated research attempt; that attempt is not peer-reviewed evidence of settlement. The statement covers the original source questions rather than only the title’s subcase.

Short English statement

Identify significant analytic or arithmetic properties of L-series attached to mixed mock modular forms.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Proved a Mellin identity in a checked growth half-plane and exposed completion and multiplicativity requirements. No universal mixed-mock L-function continuation, functional equation or Euler product is proved.
Sources and literature check.
Read the supplied complete definition, catalogue wording, scope qualifications and source list for this canonical ID. The original-source citations are inherited as supplied references; no fresh primary-source or novelty search was performed for this attempt. No current open-status claim is inferred from this calculation.
A Mellin representation under explicit growth hypotheses.
Let $f(\tau)=\sum_{n\ge1}a_ne^{2\pi in\tau}$, with $|a_n|\le Cn^A$, $A\ge0$. For $\operatorname{Re}s>A+1$, absolute convergence permits termwise integration and the substitution $u=2\pi nt$: \[\int_0^\infty f(it)t^{s-1}dt=(2\pi)^{-s}\Gamma(s)\sum_{n\ge1}a_nn^{-s}. \] Indeed the sum of the absolute integrals is bounded by a constant times $\sum n^{A-\operatorname{Re}s}$. This gives an analytic starting half-plane for an L-series, whether or not the function is modular.
A functional-equation obstruction.
Suppose a completion $\widehat f=f+R$ has a prescribed transformation under $t\mapsto1/t$. Splitting its Mellin integral at one and changing variables in the lower interval gives a relation involving both $f$ and $R$. Dropping $R$ discards the Mellin transform of the nonholomorphic correction. Without proving that correction vanishes or has a controlled continuation, a functional equation for the holomorphic-coefficient Dirichlet series does not follow. Mixed products can also alter growth at cusps, so the polynomial bound above must be checked rather than assumed universally.
Arithmetic check and gap.
An Euler product with normalized coefficient $a_1=1$ forces multiplicativity $a_{mn}=a_ma_n$ for coprime $m,n$, by multiplying its prime-power factors. Neither the Mellin identity nor the existence of a modular completion proves that condition. The first missing step for a concrete mixed mock form is to calculate its completion/cusp corrections and coefficient arithmetic. This elementary framework supplies no universal Euler product or functional equation and does not pretend the source asked for one fixed theorem.
Verification and adversarial audit.
Checked absolute Fubini bound, gamma scaling, convergence half-plane, nonholomorphic correction and the necessary Euler-product coefficient relation. This was an LLM adversarial audit, not independent human verification; no theorem prover was used. Prompts A then B applied to this ID: Checked absolute Fubini bound, gamma scaling, convergence half-plane, nonholomorphic correction and the necessary Euler-product coefficient relation. Scope remains limited: For a specified mixed mock form, compute cusp growth and nonholomorphic Mellin corrections and verify any arithmetic multiplicativity.

Coordinator checked absolute Mellin interchange in the stated growth half-plane and the coprime multiplicativity consequence of an Euler product. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 and local shell used to read supplied files and serialize this manuscript; mathematics derived symbolically in the text, with no numerical experiment or formal proof assistant. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. For a specified mixed mock form, compute cusp growth and nonholomorphic Mellin corrections and verify any arithmetic multiplicativity.

Completion Estimate

COMPLETION: 0% The calculation is an elementary benchmark or conditional reduction; it establishes no claimed new portion of the complete source target.

Sources

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