ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The functional equation gives $ζ (0) = - 1 / 2$ without substituting into a zero-times-pole expression

Example

$ζ (0) = - \frac{1}{2} .$

Facts & Assumptions

Given: The classical functional equation and the pole at $1$ .

[L1]

Zeta satisfies $ζ (s) = 2^{s} π^{s - 1} \sin (π s / 2) Γ (1 - s) ζ (1 - s)$ (The Riemann zeta function satisfies the classical sine-gamma functional equation).

[L2]

Zeta has a simple residue-one pole at $1$ (The Riemann zeta function extends meromorphically to the complex plane with its only pole at $1$ ).

Verification

technique · direct

1.1

Let $s \to 0$ in [L1]. Then $2^{s} \to 1$ , $π^{s - 1} \to π^{- 1}$ , $\sin (π s / 2) \sim π s / 2$ , and $Γ (1 - s) \to 1$ . Also [L2] gives $ζ (1 - s) \sim - 1 / s$ .

L1L2givenalgebra

2.1

Multiplying the limits from step 1.1 yields $ζ (0) = \lim_{s \to 0} 2^{s} π^{s - 1} \sin (π s / 2) Γ (1 - s) ζ (1 - s) = - \frac{1}{2} .$

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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 11 §3 (standard reference, not scraped)

The functional equation gives ζ(0)=−1/2ζ(0)=−1/2 without substituting into a zero-times-pole expression