False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: the Gamma recurrence and factorial values characterize Gamma

Statement

False claim: If a meromorphic function $F$ satisfies $F (z + 1) = z F (z)$ and $F (n + 1) = n!$ for every integer $n \geq 0$ , then $F = Γ$ .

Facts & Assumptions

Given: The Gamma recurrence and factorial values.

[L1]

Gamma satisfies $F (z + 1) = z F (z)$ on its domain (The Gamma functional equation).

[L2]

Gamma satisfies $Γ (n + 1) = n!$ for integers $n \geq 0$ (Gamma at the positive integers).

Refutation

technique · direct

1.1

Define $F (z) : = Γ (z) e^{\sin (2 π z)}$ . Since $\sin (2 π (z + 1)) = \sin (2 π z)$ , the exponential factor is $1$ -periodic. Hence [L1] gives $F (z + 1) = z F (z)$ .

givenL1construct

2.1

For every integer $n$ , $\sin (2 π n) = 0$ , so $e^{\sin (2 π n)} = 1$ . Thus [L2] gives $F (n + 1) = Γ (n + 1) = n!$ . But $e^{\sin (2 π z)} \neq 1$ identically, so $F \neq Γ$ . Therefore the stated data do not characterize Gamma.

step 1.1L2algebra∎

Depends on

Used by

A periodic perturbation preserves the Gamma recurrence and factorial values Counterexample

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 7 §4 (standard reference, not scraped)