 Mellin representation of the completed Riemann zeta function

ID: mellin-representation-of-the-completed-riemann-zeta-function

For

Θ (u) = \sum_{n \in Z} e^{- π n^{2} u}

and initially

ℜ s > 1

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

Θ (u) = u^{- 1/2} Θ (1/ u)

. Splitting the integral at one therefore continues it meromorphically and makes its invariance under

s \mapsto 1 - s

visible.

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