ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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

Example

Assume countable choice. 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: Countable choice, the Mellin representation and the completed functional equation.

[A1]

Countable choice is The Axiom of Countable Choice ( ${A C}_{ω}$ ); it supplies the premise of the theta transformation used to obtain [L2].

[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]

Under countable choice, the completed-function theorem supplies the split formula displayed in the example (The completed zeta function satisfies $Λ (s) = Λ (1 - s)$ ).

Verification

technique · direct

1.1A1L1given

Start from [L1] and split the integral at $1$ . The piece on $(0, 1)$ is exactly where the theta transformation is used under [A1] in the proof of the completed functional equation.

2.1A1step 1.1L2algebra∎

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 $1 / (s (s - 1))$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

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