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

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Splitting the theta Mellin integral at $1$ isolates the two polar terms of completed zeta

Example

For the completed zeta function,

$Λ (s) = \frac{1}{s (s - 1)} + \frac{1}{2} \int_{1}^{\infty} (θ (t) - 1) (t^{s / 2 - 1} + t^{- s / 2 - 1 / 2}) d t .$

Facts & Assumptions

Given: The Mellin representation and the completed functional equation.

[L1]

On $Re s > 1$ , $Λ (s) = \frac{1}{2} \int_{0}^{\infty} (θ (t) - 1) t^{s / 2 - 1} d t$ (The completed zeta function has its Mellin-theta integral representation on $Re s > 1$ ).

[L2]

The completed-function theorem supplies the split formula displayed in the statement (The completed zeta function satisfies $Λ (s) = Λ (1 - s)$ ).

Verification

technique · direct

1.1

Start from [L1] and split the integral at $1$ . The piece on $(0, 1)$ is exactly where the theta transformation is used in the proof of the completed functional equation.

L1given

2.1

The resulting rewritten form is the symmetric identity recorded in [L2], $Λ (s) = \frac{1}{s (s - 1)} + \frac{1}{2} \int_{1}^{\infty} (θ (t) - 1) (t^{s / 2 - 1} + t^{- s / 2 - 1 / 2}) d t,$ so the two polar terms are isolated explicitly in the factor $1 / (s (s - 1))$ .

step 1.1L2algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Elias M. Stein and Rami Shakarchi, Complex Analysis, Theorem 2.3 (standard reference, not scraped)

Splitting the theta Mellin integral at 11 isolates the two polar terms of completed zeta