In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from the perspective of Tate's thesis, which I don't really have the background to appreciate yet, so I'm asking for another perspective.

The perspective I want goes something like this: the Riemann zeta function is a product of the local zeta functions of a point over every finite prime $p$, and the Gamma function should therefore be the "local zeta function of a point at the infinite prime."

Question 1: Can this intuition be made precise without the machinery of Tate's thesis? (It's okay if you think the answer is "no" as long as you convince me why I should try to understand Tate's thesis!)

Multiplying the local zeta functions for the finite and infinite primes together, we get the Xi function, which has the nice functional equation. Now, as I learned from Andreas Holmstrom's excellent answer to my question about functional equations, for the local zeta functions at finite primes the functional equation

$$\zeta(X,n-s) = \pm q^{ \frac{nE}{2} - Es} \zeta(X, s)$$

(notation explained at the Wikipedia article), which for a point is just the statement $\frac{1}{1 - p^s} = -p^{-s} \frac{1}{1 - p^{-s}}$, reflects Poincare duality in etale cohomology, and the hope is that the functional equation for the Xi function reflects Poincare duality in some conjectural "arithmetic cohomology theory" for schemes over $\mathbb{Z}$ (or do I mean $\mathbb{F}_1$?).

Question 2: Can the reflection formula for the Gamma function be interpreted as "Poincare duality" for some cohomology theory of a point "at the infinite prime"? (Is this question as difficult to answer as the more general one about arithmetic cohomology?)

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# Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from the perspective of Tate's thesis, which I don't really have the background to appreciate yet, so I'm asking for another perspective.

The perspective I want goes something like this: the Riemann zeta function is a product of the local zeta functions of a point over every finite prime $p$, and the Gamma function should therefore be the "local zeta function of a point at the infinite prime."

Question 1: Can this intuition be made precise without the machinery of Tate's thesis? (It's okay if you think the answer is "no" as long as you convince me why I should try to understand Tate's thesis!)

Multiplying the local zeta functions for the finite and infinite primes together, we get the Xi function, which has the nice functional equation. Now, as I learned from Andreas Holmstrom's excellent answer to my question about functional equations, for the local zeta functions at finite primes the functional equation reflects Poincare duality in etale cohomology, and the hope is that the functional equation for the Xi function reflects Poincare duality in some conjectural "arithmetic cohomology theory" for schemes over $\mathbb{Z}$ (or do I mean $\mathbb{F}_1$?).

Question 2: Can the reflection formula for the Gamma function be interpreted as "Poincare duality" for some cohomology theory "at the infinite prime"? (Is this question as difficult to answer as the more general one about arithmetic cohomology?)