False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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 modified function $F (s) : = e^{(s - 1 / 2)^{2}} ζ (s) .$

[L1]

Zeta satisfies the classical functional equation (The Riemann zeta function satisfies the classical sine-gamma functional equation).

Refutation

technique · direct

1.1

The factor $e^{(s - 1 / 2)^{2}}$ is entire, nonconstant, and invariant under $s \mapsto 1 - s$ because $(1 - s - 1 / 2)^{2} = (s - 1 / 2)^{2}$ . Therefore multiplying the functional equation in [L1] by this factor shows that $F$ satisfies the same functional equation as zeta.

L1givenalgebra

2.1

The function $F$ is not equal to zeta, since for example $F (2) = e^{9 / 4} ζ (2) \neq ζ (2)$ . Thus step 1.1 gives a different meromorphic function obeying the same functional equation, so that equation alone cannot characterize zeta.

step 1.1algebra∎

Depends on

The Riemann zeta function satisfies the classical sine-gamma functional equation

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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 12 §7 (standard reference, not scraped)