Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedverified 2026-09-26 (gpt-6-sol)
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  • 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.
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FALSE: the classical functional equation alone characterizes the Riemann zeta function

Statement

False claim: the classical functional equation by itself determines zeta.

Facts & Assumptions

Given: The classical homogeneous equation F(s)=2sπs−1sin⁡(πs/2)Γ(1−s)F(1−s), interpreted as an identity of meromorphic functions.

[L1]

On Re⁡s>1, ζ(s)=∑n≥1n−s (The Riemann zeta function on the half-plane Re⁡s>1).

Refutation

technique · direct
1.1givenconstructalgebra

Take the entire function F(s)=0. Both sides of the given equation are identically zero wherever its displayed coefficient is finite. At its poles the product with the zero function has the zero meromorphic extension. Thus F satisfies the equation as a meromorphic identity.

2.1L1step 1.1algebra∎

By [L1], ζ(2)=∑n≥1n−2≥1, whereas F(2)=0. Hence a meromorphic function different from zeta satisfies the equation, and the equation alone cannot determine zeta. No assertion that zeta itself satisfies the equation is needed for this counterexample.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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