TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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The completed zeta function has its Mellin-theta integral representation on $Re s > 1$

Statement

If $Re s > 1$ , then

$π^{- s / 2} Γ (s / 2) ζ (s) = \frac{1}{2} \int_{0}^{\infty} (θ (t) - 1) t^{s / 2 - 1} d t .$

Facts & Assumptions

Given: A complex number $s$ with $Re s > 1$ .

[L1]

On $Re s > 1$ , $ζ (s) = \sum_{n \geq 1} n^{- s}$ (The Riemann zeta function on the half-plane $Re s > 1$ ).

[L2]

For $t > 0$ , $θ (t) - 1 = 2 \sum_{n \geq 1} e^{- π n^{2} t}$ (The Jacobi theta function $θ (t) = \sum_{n \in Z} e^{- π n^{2} t}$ for $t > 0$ ).

[L3]

On $Re z > 0$ , $Γ (z) = \int_{0}^{\infty} e^{- u} u^{z - 1} d u$ (Euler's Gamma function on the right half-plane).

[L4]

Tonelli's theorem permits swapping a nonnegative sum and integral on a sigma-finite product (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

Proof

technique · direct

1.1

Write $σ : = Re s$ . By [L2], $\frac{1}{2} \int_{0}^{\infty} (θ (t) - 1) t^{s / 2 - 1} d t = \sum_{n \geq 1} \int_{0}^{\infty} e^{- π n^{2} t} t^{s / 2 - 1} d t,$ provided the interchange is justified. Since $∣ t^{s / 2 - 1} ∣ = t^{σ / 2 - 1}$ and $σ > 1$ , the summands are absolutely integrable and nonnegative after taking absolute values, so [L4] applies to the absolute-value kernel.

givenL2L4algebra

1.2

For each $n \geq 1$ , substitute $u = π n^{2} t$ . Then $\int_{0}^{\infty} e^{- π n^{2} t} t^{s / 2 - 1} d t = π^{- s / 2} n^{- s} \int_{0}^{\infty} e^{- u} u^{s / 2 - 1} d u = π^{- s / 2} Γ (s / 2) n^{- s}$ by [L3].

L3algebra

2.1

Summing the identity of step 1.2 over $n$ and using [L1] yields $\frac{1}{2} \int_{0}^{\infty} (θ (t) - 1) t^{s / 2 - 1} d t = π^{- s / 2} Γ (s / 2) \sum_{n \geq 1} n^{- s} = π^{- s / 2} Γ (s / 2) ζ (s) .$ This is the claimed Mellin representation.

step 1.1step 1.2L1algebra∎

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Sources

Elias M. Stein and Rami Shakarchi, Complex Analysis, Theorem 2.2 (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 12 §7 (standard reference, not scraped)

The completed zeta function has its Mellin-theta integral representation on Re⁡s>1Res>1