CorollaryStatement: Literature-sourcedProof: 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.

Gamma has no zeros

Statement

Gamma has no zeros on $C$ , and $1 / Γ$ has simple zeros exactly at $0, - 1, - 2, \dots$ .

Facts & Assumptions

Given: The reciprocal-Gamma product.

[L1]

One has $1 / Γ (z) = z e^{γ z} \prod_{n \geq 1} (1 + z / n) e^{- z / n}$ (The Weierstrass product for reciprocal Gamma).

Proof

technique · direct

1.1

In [L1], the exponential factors never vanish, while the factor $z$ gives a simple zero at $0$ and the factor $1 + z / n$ gives a simple zero at $z = - n$ for each $n \geq 1$ . There are no other zeros.

L1given

2.1

Therefore $1 / Γ$ vanishes exactly at the nonpositive integers, and Gamma has poles there and no zeros anywhere.

step 1.1algebra∎

Depends on

The Weierstrass product for reciprocal Gamma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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 6 §3 (standard reference, not scraped)