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.]

— Riemann’s proof —

Applying the Poisson summation formula (2) for $k_\infty$ to the Schwartz function (8), we see that the theta function

$\displaystyle\Theta_\infty( x_\infty ) := \sum_{n \in {\Bbb Z}} g_\infty( n x_\infty ) = 1 + 2 \sum_{n=1}^\infty e^{-\pi n^2 |x_\infty|_\infty^2}$ (10)

obeys the functional equation

$\displaystyle\Theta_\infty( x_\infty ) = \frac{1}{|x_\infty|_\infty} \Theta_\infty( \frac{1}{x_\infty} )$ (11)

for $x_\infty \in k_\infty^\times := k_\infty \backslash \{0\}$.  In particular, since $\Theta_\infty(x_\infty)-1$ is rapidly decreasing as $x_\infty \to \infty$, we see that $\Theta_\infty(x_\infty) - 1/x_\infty$ is rapidly decreasing as $x_\infty \to 0$.

Formally, we can take Mellin transforms of (11) and conclude that

$\displaystyle\int_{k_\infty^\times} \Theta_\infty(x_\infty) |x_\infty|_\infty^s d^\times x_\infty = \int_{k_\infty^\times} \Theta_\infty(x_\infty) |x_\infty|_\infty^{1-s} d^\times x_\infty$ (12)

for any s, where $d^\times x_\infty := \frac{dx_\infty}{|x_\infty|_\infty}$ is the standard multiplicative Haar measure on $k_\infty^\times$.  This does not make rigorous sense, because the integrands here diverge at 0 and at infinity (which is ultimately due to the poles of the Riemann Xi function at s=0 and s=1), but let us forge ahead regardless.  By making the change of variables $y := \pi n^2 t^2$ and using (3), (4), we see that

$\displaystyle \int_{k_\infty}e^{-\pi n^2 x_\infty^2} |x_\infty|_\infty^s d^\times x_\infty = \Gamma_\infty(s) n^{-s}$ (13)

and so from (10) and (1) we formally have

$\displaystyle \int_{k_\infty} \Theta_\infty(x_\infty) |x_\infty|_\infty^s d^\times x_\infty = \int_{k_\infty} |x_\infty|^s d^\times x_\infty + 2\Gamma_\infty(s) \zeta(s)$. (14)

If we casually discard the divergent integral $\int_{k_\infty} |x_\infty|^s d^\times x_\infty$ and apply (12), we formally obtain the functional equation (2).

Of course, the above computations were totally formal in nature.  Nevertheless it is possible to make the argument rigorous.  For instance, when $\hbox{Re}(s) > 1$, we have a rigorous version of (14), namely

$\displaystyle \int_{k^\infty} (\Theta_\infty(x_\infty)-1) |x_\infty|_\infty^s d^\times x_\infty = 2 \Gamma_\infty(s) \zeta(s)$, (15)

which can be deduced from (13) and Fubini’s theorem (or by dominated convergence).  Using (11) and a little undergraduate calculus, we can rewrite the left-hand side of (15) as

$\displaystyle 2[ \int_1^\infty (\Theta_\infty(t)-1) (t^s + t^{1-s}) \frac{dt}{t} - \frac{1}{s} - \frac{1}{1-s}]$. (16)

Observe that this expression extends meromorphically to all of s and can thus be taken as a definition of $\Xi(s)$ for all $s \neq 0, 1$, and the functional equation (2) is then manifestly obvious.

Here is a slightly different way to view the above computations.  Since the Gaussian (8) is its own Fourier transform, we see for every $t > 0$ that the Fourier transform of $e^{-\pi t^2 |x|_\infty^2}$ is $\frac{1}{t} e^{-\pi |x|_\infty^2/t^2}$.  Integrating this fact against $|t|^s d^\times t$ on $k_\infty^\times$ using (13), we obtain (formally) at least that the Fourier transform of $\Gamma_\infty(s) |x|_\infty^{-s} d^\times x$ is $\Gamma_\infty(1-s) |x|_\infty^{1-s} d^\times x$.  (Note from scaling considerations it is formally clear that the Fourier transform of $|x|_\infty^{-s} d^\times x$ must be some sort of constant multiple of $|x|_\infty^{1-s} d^\times x$; the Gamma factors can thus be viewed as the normalisation of these multiplicative characters that is compatible with the Fourier transform.) Formally applying the Poisson summation formula (4) to this, and casually discarding the singular terms at the origin, we obtain (2).  One can make the above computations rigorous using the theory of distributions, and by using Gaussians to regularise the various integrals and summations appearing here, in which case the computations become essentially equivalent to the previous ones.

The above “Archimedean” Fourier analysis on $k_\infty = {\Bbb R}$ has analogues in the p-adic completions $k_p = {\Bbb Q}_p$ of the rationals $k = {\Bbb Q}$.  (This is analogous to my discussion of dyadic models, although the p-adics ${\Bbb Z}_p, {\Bbb Q}_p$ are still characteristic 0 and are thus not as dyadic as their function field cousins ${\Bbb F}_p[[t]]$, ${\Bbb F}_p((t))$.)

Recall that the reals $k_\infty = {\Bbb R}$ are the metric completion of the rationals $k = {\Bbb Q}$ with respect to the metric arising from the usual Archimedean absolute value $x \mapsto |x|_\infty$.  This absolute value obeys the following basic properties:

1. Positivity: we have $|x| \geq 0$ for all x, with equality if and only if x=0.
2. Multiplicativity: we have $|xy| = |x| |y|$ for all x, y.
3. Triangle inequality: We have $|x+y| \leq |x| + |y|$ for all x, y.

A function from k to ${}[0,+\infty)$ with the above three properties is known as an absolute value (or valuation) on k.  In addition to the Archimedean absolute value, each prime p defines a p-adic absolute value $x \mapsto |x|_p$ on k, defined by the formula $|x|_p := p^{-n}$, where n is the number of times p divides x.  (This number could be negative if the denominator of the rational number x contains factors of p.)  Equivalently, $|x|_p$ is the unique valuation such that $|p|_p = 1/p$ and $|n|_p = 1$ whenever n is an integer coprime to p.

One easily verifies that $|x|_p$ is an absolute value; in fact it not only obeys the triangle inequality, but also the ultra-triangle inequality $|x+y| \leq \max(|x|, |y|)$, making the p-adic absolute value a non-Archimedean absolute value.

A classical theorem of Ostrowski asserts that the Archimedean absolute value $x \mapsto |x|_\infty$ and the p-adic absolute values $x \mapsto |x|_p$ are in fact the only absolute values on the rationals k, up to the renormalisation of replacing an absolute value |x| with a power $|x|^\alpha$.  If we define a place to be an absolute value up to renormalisation, we thus see that the rationals k have one Archimedean (or infinite) place $\infty$, together with one non-Archimedean (or finite) place p for every prime.

One could have set $|p|_p$ to some other value between 0 and 1 than 1/p (thus replacing $|x|_p$ with some power $|x|_p^\alpha$) and still get an absolute value; but this normalisation is natural because it allows one to write the fundamental theorem of arithmetic in the appealing form

$\displaystyle \prod_\nu |x|_\nu = 1$ for all $x \in k^\times$ (17)

where $\nu$ ranges over all places, and $k^\times := k \backslash \{0\}$ is the multiplicative group of k.

If one takes the metric completion of the rationals $k = {\Bbb Q}$ using a p-adic absolute value $|x|_p$ rather than the Archimedean one, one obtains the p-adic field $k_p = {\Bbb Q}_p$.  One can view this field as a kind of inverted version of the real field ${\Bbb R}$, in which p has been inverted to be small rather than large.  Some illustrations of this inversion:

1. In $k_\infty$, the sequence $p^n$ goes to infinity as $n \to +\infty$ and goes to zero as $n \to -\infty$; in $k_p$, it is the other way around.
2. Elements of $k_\infty$ can be expressed base p as strings of digits that need not terminate to the right of the decimal point, but must terminate to the left.  In $k_p$, it is the other way around.  (The famous ambiguity $0.999\ldots=1.000\ldots$ in $k_\infty$ does not occur in the p-adic field $k_p$, because the latter has the topology of a Cantor space rather than a continuum.)
3. In $k_\infty$, the integers ${\Bbb Z}$ is closed and forms a discrete cocompact additive subgroup.  In $k_p$, the integers are not closed, but their closure ${\mathcal O}_p = {\Bbb Z}_p$ (the ring of p-adic integers) forms a compact codiscrete additive subgroup.

Despite this inversion, we can obtain analogues of most of the additive and multiplicative Fourier analytic computations of the previous section for the p-adics.

Let’s first begin with the additive Fourier structure.  By the theory of Haar measures, there is a unique translation-invariant measure $dx_p$ on $k_p = {\Bbb Q}_p$ which assigns a unit mass to the compact codiscrete subgroup ${\mathcal O}_p = {\Bbb Z}_p$.  One can check that this measure interacts with dilations in the expected manner, thus

$\displaystyle \int_{k_p} f(tx_p)\ dx_p = \frac{1}{|t|_p} \int_{k_p} f(x_p)\ dx_p$ (18)

for all absolutely integrable f and all invertible $t \in k_p^\times := k_p \backslash \{0\}$.

Just as $k_\infty$ has a standard character $e_\infty: x \mapsto e^{2\pi i x}$, we can define a standard character $e_p: k_p \to S^1$ as the unique character (i.e. continuous homomorphism from $k_p$ to $S^1$) such that $e_p(p^n) = e^{2\pi i p^n}$ for all integers n (in particular, $e_p$ is trivial on the integers, just as $e_\infty$ is).  One easily verifies that this is indeed a character.   From this and the additive Haar measure $dx_p$, we can now define the p-adic Fourier transform

$\displaystyle \hat f(\xi_p) := \int_{k_p} e_p( - x_p \xi_p ) f(x_p)\ dx_p$ (19)

for reasonable (e.g. absolutely integrable) f, and it is a routine matter to verify all the usual Fourier-analytic identities for this transform (or one can appeal to the general theory of Fourier analysis on locally compact abelian groups).

In $k_\infty$, we have the Gaussian function (8), which is its own Fourier transform.  In $k_p$, the analogous Gaussian function $g_p: k_p \to {\Bbb C}$ is given by the formula

$\displaystyle g_p := 1_{{\mathcal O}_p}$, (20)

i.e. the p-adic Gaussian is just the indicator function of the p-adic integers.  One easily verifies that this function is also its own Fourier transform.

Now we turn to the multiplicative Fourier theory for $k_p$.  The natural multiplicative Haar measure $d^\times x_p$ on $k_p^\times$ is given by the formula $d^\times x_p := \frac{p}{p-1} \frac{dx_p}{|x_p|_p}$; the normalisation factor $\frac{p}{p-1}$ is natural in order for the group of units ${\mathcal O}_p^\times = {\Bbb Z}_p \backslash p {\Bbb Z}_p$ to have unit mass.

In $k_\infty$, we see from (13) that the Gamma factor at infinity can be expressed (for $\hbox{Re}(s) > 1$) as the Mellin transform of the Gaussian:

$\displaystyle \Gamma_\infty(s) = \int_{k_\infty^\times} g_\infty(x_\infty) |x_\infty|_\infty^s d^\times x_\infty$. (21)

In analogy with this, we can define the Gamma factor at p by the formula

$\displaystyle \Gamma_p(s) = \int_{k_p^\times} g_p(x_p) |x_p|_p^s d^\times x_p$. (22)

Due to the simple and explicit nature of all the expressions on the right-hand side, it is a straightforward matter to compute this factor explicitly; it becomes

$\displaystyle \Gamma_p(s) = (1 - p^{-s})^{-1}$ (23)

for $\hbox{Re}(s) > 1$, at least; of course, one can then extend $\Gamma_p$ meromorphically in the obvious manner.

In $k_\infty$, we showed (formally, at least) that $\Gamma_\infty(s) |x|_\infty^s$ and $\Gamma_\infty(1-s) |x|_\infty^{1-s}$ were Fourier transforms of each other.  One can similarly show that $\Gamma_p(s) |x|_p^s$ and $\Gamma_p(1-s) |x|_p^{1-s}$ are Fourier transforms of each other in $k_p$.

On the other hand, there is no obvious analogue of the Poisson summation formula manipulations for $k_p$, because (unlike $k_\infty$), $k_p$ lacks a discrete cocompact subgroup.

— Tate’s proof —

We have just performed some “local” additive and multiplicative Fourier analysis at a single place.  (This use of “local” may seem unrelated to the topological or analytical notion of “local”, as in “in the vicinity of a single point”, but it is actually much the same concept; compare for instance the formal power series in p for a p-adic number with the Taylor series expansion in $t-t_0$ of a function f(t) around a point $t_0$.  Indeed one can view local analysis at a place p as being the analysis of the integers or rationals when p is “close to zero”; one can make this precise using the language of schemes, but we will not do so here.)

In his famous thesis, Tate observed that all these local Fourier-analytic computations could be unified into a single global Fourier-analytic computation, using the languge of the adele ring ${\Bbb A}$.  This ring is the set of all tuples $x = (x_\nu)_\nu$, where $\nu$ ranges over places and $x_\nu \in k_\nu$, and furthermore all but finitely many of the $x_\nu$ lie in their associated ring of integers ${\mathcal O}_\nu$.  (Equivalently, the adele ring is the tensor product of the rationals $k = {\Bbb Q}$ with the ring of integral adeles ${\Bbb R} \times \prod_p {\Bbb Z}_p$.)  This restriction that the $x_\nu$ consists mostly of integers in $k_\nu$ is important for a large variety of analytic and algebraic reasons; for instance, it keeps the adele ring $\sigma$-compact.

Many of the structures and objects on the local fields $k_\nu$ can be multiplied together to form corresponding global structures on the adele ring.  For instance:

1. The commutative ring structures on the $k_\nu$ multiply together to give a commutative ring structure on ${\Bbb A}$.
2. The locally compact Hausdorff structures on the $k_\nu$ multiply together to give a locally compact Hausdorff structure on ${\Bbb A}$.
3. The local additive Haar measures $dx_\nu$ on the $k_\nu$ multiply together to give a global additive Haar measure $dx$ on ${\Bbb A}$.
4. The local characters $e_\nu: k_\nu \to S^1$ on the $k_\nu$ multiply together to give a global character $e: {\Bbb A} \to S^1$ (here it is essential that most components of an adele are integers, and so are trivial with respect to their local character).
5. The local additive Fourier transforms on the $k_\nu$ then multiply to form a global additive Fourier transform on ${\Bbb A}$, defined as $\hat f(\xi) := \int_{\Bbb A} e(-x \xi) f(x)\ dx$ for reasonable f.
6. The local absolute values $x_\nu \mapsto |x_\nu|_\nu$ on $k_\nu$ multiply to form a global absolute value $x \mapsto |x|$ on ${\Bbb A}$, though with the important caveat that |x| can vanish for non-zero x (indeed, a simple calculation using Euler’s observation $\prod_p (1-\frac{1}{p})=0$ shows that almost every x does this, with respect to additive Haar measure).  The x for which |x| is non-zero are invertible and known as ideles, and form a multiplicative group ${\Bbb A}^\times$; the ideles have measure zero inside the adeles.
7. The local gaussians $g_\nu: k_\nu \to {\Bbb C}$ multiply together to form a global gaussian $g: {\Bbb A} \to {\Bbb C}$, which is its own Fourier transform.
8. The embeddings $k \subset k_\nu$ at each place $\nu$ multiply together to form a diagonal embedding $k \subset {\Bbb A}$.  This embedding is both discrete (by the fundamental theorem of arithmetic) and cocompact (this is basically because the integers are cocompact in the adelic integers).
9. The local multiplicative Haar measures $d^\times x_\nu$ multiply together to form a global multiplicative Haar measure $d^\times x$, though one should caution that this measure is supported on the ideles ${\Bbb A}^\times$ rather than the adeles ${\Bbb A}$.
10. The local Gamma factors $\Gamma_\nu(s)$ for each place $\nu$ multiply together to form the Riemann Xi function (3) (for $\hbox{Re}(s)>1$ at least), thanks to the Euler product formula $\zeta(s) = \prod_p (1-p^{-s})^{-1}$.

Recall that the local Gamma factors were the local Mellin transforms of the local Gaussians.  Multiplying this together, we see that the Riemann Xi function is the global Mellin transform of the global Gaussian:

$\displaystyle \Xi(s) = \int_{{\Bbb A}^\times} g(x) |x|^s d^\times x$. (24)

Our derivation of (24) used the Euler product formula.  Another way to establish (24) using the original form (1) of the zeta function is to observe (thanks to the fundamental theorem of arithmetic) that the set $J := {\Bbb R}^+ \times \prod_p {\mathcal O}_p^\times$ is a fundamental domain for the action of $k^\times$ on ${\Bbb A}^\times$, thus

$\displaystyle {\Bbb A}^\times = \biguplus_{a\in k^\times} a \cdot J$. (25)

Partitioning (24) using (25) and then using (13) and (1) one can give an alternate derivation of (24).  (The two derivations are ultimately the same, of course, since the Euler product formula is itself essentially a restatement of the fundamental theorem of arithmetic.)

In his thesis, Tate established the Poisson summation formula (9) for the adeles for all sufficiently nice f (e.g. any f in the Schwartz-Bruhat class would do).  Applying this to the global gaussian g, we conclude that the global Theta function $\Theta(x) := \sum_{a \in k} g(ax)$ obeys the functional equation

$\displaystyle \Theta(x) = \frac{1}{|x|} \Theta(\frac{1}{x})$ (26)

for all ideles t.  This formally implies that

$\displaystyle \int_J \Theta(x) |x|^s d^\times x = \int_J \Theta(x) |x|^{1-s} d^\times x$ (27),

which on applying (25) and (24) (and casually discarding the singular contributions of $a=0$) yields the functional equation (2).  One can make this formal computation rigorous in exactly the same way that Riemann’s proof was made rigorous in previous sections.

Recall that Riemann’s proof could also be established by inspecting the Fourier transforms of $\displaystyle \Gamma_\infty(s) |x_\infty|_\infty^s d^\times x_\infty$.  A similar approach can work here.  If we (very formally!) apply the Poisson summation formula (9) to the measure $\displaystyle 1_J(x) |x|^s\ d^\times x$, one obtains

$\displaystyle 1 = \sum_{a \in k} \int_J e(-ax) |x|^s\ d^\times x$. (28)

Unpacking this summation using (25) (and (17)), and casually discarding the a=0 term, we formally conclude that

$\displaystyle 1 = \int_{{\mathbb A}^\times} e(-x) |x|^s\ d^\times x$. (29)

Rescaling this, we formally conclude that the Fourier transform of $|x|^s\ d^\times x$ is $|x|^{1-s}\ d^\times x$.  Inserting this into (24), the functional equation (2) formally follows from Parseval’s theorem; alternatively, one can derive it by multiplying together all the local facts that the Fourier transform of $\Gamma_\nu(s) |x_\nu|_\nu^s\ d^\times x_\nu$ in $k_\nu$ is $\Gamma_\nu(1-s) |x_\nu|_\nu^{1-s}\ d^\times x_\nu$.  These arguments can be made rigorous using the theory of distributions (and a lot of care), but we will not do so here.