Alphabeta MathHow 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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Gamma has no zeros
Statement
Gamma has no zeros on , and has simple zeros exactly at .
Facts & Assumptions
Given: The reciprocal-Gamma product.
One has (The Weierstrass product for reciprocal Gamma).
Proof
In [L1], the exponential factors never vanish, while the factor gives a simple zero at and the factor gives a simple zero at for each . There are no other zeros.
Therefore vanishes exactly at the nonpositive integers, and Gamma has poles there and no zeros anywhere.
Depends on
Used by
- K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing Lemma
- Residues in the von Mangoldt contour shift Lemma
- The only zeros of zeta on the nonpositive real axis are the negative even integers, and every other zero lies in the open critical strip Theorem
- The Riemann xi function has its genus-one Hadamard product over the nontrivial zeros of zeta Theorem
- The Riemann zeta function extends meromorphically to the complex plane with its only pole at 1 Theorem
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)