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The Riemann zeta function $\zeta(s)$, defined for $\hbox{Re}(s) > 1$ by the formula

$\displaystyle \zeta(s) := \sum_{n \in {\Bbb N}} \frac{1}{n^s}$ (1)

where ${\Bbb N} = \{1,2,\ldots\}$ are the natural numbers, and extended meromorphically to other values of s by analytic continuation, obeys the remarkable functional equation

$\displaystyle \Xi(s) = \Xi(1-s)$ (2)

where

$\displaystyle \Xi(s) := \Gamma_\infty(s) \zeta(s)$ (3)

is the Riemann Xi function,

$\displaystyle \Gamma_\infty(s) := \pi^{-s/2} \Gamma(s/2)$ (4)

is the Gamma factor at infinity, and the Gamma function $\Gamma(s)$ is defined for $\hbox{Re}(s) > 1$ by

$\displaystyle \Gamma(s) := \int_0^\infty e^{-t} t^s\ \frac{dt}{t}$ (5)

and extended meromorphically to other values of s by analytic continuation.

There are many proofs known of the functional equation (2).  One of them (dating back to Riemann himself) relies on the Poisson summation formula

$\displaystyle \sum_{a \in {\Bbb Z}} f_\infty(a t_\infty) = \frac{1}{|t|_\infty} \sum_{a \in {\Bbb Z}} \hat f_\infty(a/t_\infty)$ (6)

for the reals $k_\infty := {\Bbb R}$ and $t \in k_\infty^*$, where $f$ is a Schwartz function, $|t|_\infty := |t|$ is the usual Archimedean absolute value on $k_\infty$, and

$\displaystyle \hat f_\infty(\xi_\infty) := \int_{k_\infty} e_\infty(-x_\infty \xi_\infty) f_\infty(x_\infty)\ dx_\infty$ (7)

is the Fourier transform on $k_\infty$, with $e_\infty(x_\infty) := e^{2\pi i x_\infty}$ being the standard character $e_\infty: k_\infty \to S^1$ on $k_\infty$.  (The reason for this rather strange notation for the real line and its associated structures will be made clearer shortly.)  Applying this formula to the (Archimedean) Gaussian function

$\displaystyle g_\infty(x_\infty) := e^{-\pi |x_\infty|^2}$, (8)

which is its own (additive) Fourier transform, and then applying the multiplicative Fourier transform (i.e. the Mellin transform), one soon obtains (2).  (Riemann also had another proof of the functional equation relying primarily on contour integration, which I will not discuss here.)  One can “clean up” this proof a bit by replacing the Gaussian by a Dirac delta function, although one now has to work formally and “renormalise” by throwing away some infinite terms.  (One can use the theory of distributions to make this latter approach rigorous, but I will not discuss this here.)  Note how this proof combines the additive Fourier transform with the multiplicative Fourier transform.  [Continuing with this theme, the Gamma function (5) is an inner product between an additive character $e^{-t}$ and a multiplicative character $t^s$, and the zeta function (1) can be viewed both additively, as a sum over n, or multiplicatively, as an Euler product.]

In the famous thesis of Tate, the above argument was reinterpreted using the language of the adele ring ${\Bbb A}$, with the Poisson summation formula (4) on $k_\infty$ replaced by the Poisson summation formula

$\displaystyle \sum_{a \in k} f(a t) = \sum_{a \in k} \hat f(t/a)$ (9)

on ${\Bbb A}$, where $k = {\Bbb Q}$ is the rationals, $t \in {\Bbb A}$, and f is now a Schwartz-Bruhat function on ${\Bbb A}$.  Applying this formula to the adelic (or global) Gaussian function $g(x) := g_\infty(x_\infty) \prod_p 1_{{\mathbb Z}_p}(x_p)$, which is its own Fourier transform, and then using the adelic Mellin transform, one again obtains (2).  Again, the proof can be cleaned up by replacing the Gaussian with a Dirac mass, at the cost of making the computations formal (or requiring the theory of distributions).

In this post I will write down both Riemann’s proof and Tate’s proof together (but omitting some technical details), to emphasise the fact that they are, in some sense, the same proof.  However, Tate’s proof gives a high-level clarity to the situation (in particular, explaining more adequately why the Gamma factor at infinity (4) fits seamlessly with the Riemann zeta function (1) to form the Xi function (2)), and allows one to generalise the functional equation relatively painlessly to other zeta-functions and L-functions, such as Dedekind zeta functions and Hecke L-functions.

[Note: the material here is very standard in modern algebraic number theory; the post here is partially for my own benefit, as most treatments of this topic in the literature tend to operate in far higher levels of generality than I would prefer.]