
There are lots of different things called "zeta functions" in mathematics and physics. The granddaddy of them all is the Riemann zeta function:
ζ(s) = 1/1^{s} + 1/2^{s} + 1/3^{s} + 1/4^{s} + ....
This is deeply related to prime numbers, thanks to Euler's product formula
ζ(s) = ∏ 1/(1  p^{s})
where we take a product over all primes. This formula is fun to prove: just use the geometric series to expand each factor, multiply them out and see what happens!
Using this, Riemann and von Mangoldt derived an explicit formula for how many primes are less than a given number as a sum over the zeros of the Riemann zeta function. Instead of showing you this formula, I'll just urge you to watch a movie of how it works:
1) Matthew Watkins, Animation: the prime counting function π(x), http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm
Thanks to this formula, information about the Riemann zeta function is secretly information about the distribution of primes!
For example, the Riemann Hypothesis says that when we analytically continue the zeta function to the complex plane, the only zeros occur at negative even integers and numbers with real part equal to 1/2. And, knowing this would be equivalent to knowing that the number of primes less than x differs from
x Li(x) = ∫ dt/ln t 2by less than some constant times ln(x) √x. Everyone feels sure these facts are true. But, despite over a century of hard work and a milliondollar prize offered by the Clay Mathematics Institute, nobody has come close to proving them!
It's known that apart from the negative even integers, the only place the Riemann ζ function can vanish is in the strip where
0 < Re(s) < 1
But, nobody has been able to show that all the zeros in this "critical strip" lie on the line
Re(s) = 1/2
Of course, this can be checked in special cases. The current record may belong to Xavier Gourdon, who on October 12th of 2004 claimed to have shown  with the help of a computer  that the first ten trillion zeros in the critical strip lie on the line Re(s) = 1/2.
2) Xavier Gourdon, Computation of zeros of the Riemann zeta function, http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html
Alas, such monster calculations don't seem helpful for proving the Riemann hypothesis. They're more useful when it comes to formulating and testing conjectures about the statistical properties of the zeros.
The most famous of these traces its way back to a teatime conversation between Hugh Montgomery and Freeman Dyson... you can read the story here:
3) K. Sabbagh, Dr. Riemann's Zeros, Atlantic Books, 2002, pp. 134136. Story about Hugh Montgomery and Freeman Dyson also available at http://www.maths.ex.ac.uk/~mwatkins/zeta/dyson.htm
The upshot is that the distribution of spacings between Riemann zeros closely mimics the spacings between eigenvalues of a large randomly chosen selfadjoint matrix.
This suggests fascinating relations between the Riemann zeta function and quantum physics. In fact, one popular dream for proving the Riemann zeta function is to find a chaotic classical system whose quantized version has energy levels related to the Riemann zeta zeros!
I would like to understand this stuff, but it all seems a bit intimidating  especially since the coolest aspects are the ones nobody understands.
Luckily, the Riemann zeta function has spawned a lot of other functions called zeta functions and Lfunctions, and many of these are simpler than the original one  or at least raise fascinating questions that are easier to solve. Many of these are listed here:
4) Matthew R. Watkins, A directory of all known zeta functions, http://www.maths.ex.ac.uk/~mwatkins/zeta/directoryofzetafunctions.htm
Matthew R. Watkins, A directory of all known Lfunctions, http://www.maths.ex.ac.uk/~mwatkins/zeta/directoryofLfunctions.htm
Lately I've been talking about zeta functions with James Dolan and also Darin Brown, whose work I mentioned last week. I feel some things are starting to make sense, so I'd like to explain them before it turns out they don't.
I'll just list some zeta functions, so you can see what we're dealing with:
A) The zeta function of a number field. A "number field" is something you get by taking the rational numbers and throwing in some algebraic numbers. One can define "algebraic integers" for any number field, and they act a lot like the ordinary integers. So, one can define a zeta function for any number field.
Technically, we do this by summing over all nonzero ideals I in the ring A of algebraic integers in our number field:
ζ(s) = ∑ I^{s}
where I is a number called the "norm" of I, which is just the cardinality of A/I. We also have an Euler product formula:ζ(s) = ∏ 1/(1  P^{s})
where we take the product over prime ideals P.
For example, if our number field is the rational numbers, its algebraic integers are the ordinary integers. So, each ideal consists of multiples of some number n = 1,2,3,..., and its norm is just n, so we get:
ζ(s) = 1/1^{s} + 1/2^{s} + 1/3^{s} + 1/4^{s} + ....
A more fun example is the number field Q[i], where we take the rational numbers and throw in a square root of 1. Here the algebraic integers are the socalled "Gaussian integers" Z[i], namely guys like
a + bi
where a and b are ordinary integers.
In this example it's easiest to work out the zeta function using the Euler product formula. If you ask one of your number theory pals about prime ideals in the Gaussian integers, they'll say:
"Well, the Gaussian integers are a principal ideal domain, so every ideal is generated by a single element. So, we can actually talk about prime numbers in the Gaussian integers. And there are 3 cases:
 INERT: An ordinary prime number of the form 4n+3 is also prime in the Gaussian integers: for example, 3.
 SPLIT: An ordinary prime numbers of the form 4n+1 is the product of two complex conjugate primes in the Gaussian integers: for example, 5 = (2+i)(2i).
 RAMIFIED: The ordinary prime 2 equals (1+i)(1i), but here the two factors give the same prime ideal, since (1i) = i (1+i), and i is invertible in the Gaussian integers.
We get all primes in the Gaussian integers this way."
So, the zeta function of the Gaussian integers goes like this:
ζ(s) = 1/(1  2^{s}) 1/(1  3^{2s}) 1/(1  5^{s}) 1/(1  5^{s}) ...
I went just far enough to show you what happens for each kind of prime. As you might expect, we get two factors for each prime that splits in two.
I should explain the other details, but number theory is best absorbed in small doses, especially if you're a physicist. The main lesson to take home is this:
A number field is like a funny sort of "branched covering space" of the set of ordinary primes. Sitting over each ordinary prime there are one or more prime ideals in our number field:
2+i 3+2i  1+i  3   7  11   GAUSSIAN INTEGERS 2i 32i 23571113 INTEGERSAnd, the ζ function records the details of how this works!
For more on this covering space philosophy see "week205" and "week213". This geometrical metaphor lies behind a lot of the really cool work on number theory.
B) The zeta function of a discrete dynamical system. A "discrete dynamical system" consists of a set X together with a onetoone and onto function
f: X → X
Here X is the set of "states" of some physical system, and f describes one step of time evolution. For each integer n we get a function
f^{ n}: X → X
Since the integers are called Z, mathematicians would call our discrete dynamical system a "Zset".
Whatever you call it, its zeta function is defined to be:
ζ(s) = ∏ 1/(1  P^{s})
where P ranges over all periodic orbits and P is the exponential of the size of this periodic orbit.
This is like an Euler product formula with the periodic orbits being the "primes". Just as every natural number can be uniquely factored into primes, every discrete dynamical system can be uniquely written as a disjoint union of periodic orbits. This explains the exponential in the definition of P above: primes like to multiply, while sizes of orbits like to add.
One nice thing about this zeta function is that when we take the disjoint union of two discrete dynamical systems, their zeta functions multiply. Another nice thing is that the zeta function of a dynamical system completely describes it up to isomorphism, at least when the set X is finite. Decategorification at work!
We can also rewrite this zeta function as a sum:
ζ(s) = ∑ I^{s}
where I ranges over all formal products of periodic orbits, and we define P_{1} ... P_{n} = P_{1} ... P_{n}.
Even better, examples A) and B) overlap. I'll explain how later, but the key is to associate to any number field and any prime p a discrete dynamical system
f: X → X
called the "Frobenius automorphism". This gives a zeta function. It works best if we take the exponential of the size of each periodic orbit using the base p instead of base e. Then, if we multiply all these zeta functions, one for each ordinary prime p, we get the zeta function of our number field!
C) The zeta function of a continuous dynamical system. Now suppose X is some topological space and we have a time evolution map
f_{ t}: X → X
for each real number t. We can define a zeta function
ζ(s) = ∏ 1/(1  P^{s})
where P ranges over all periodic orbits and P is the exponential of the "period" of P  meaning the time it takes for points on this orbit to loop around back to where they started.
A famous example is when we have a Riemannian manifold. A free particle moving around on such a space will trace out geodesics, giving us a dynamical system. The analogue of primes are now "prime geodesics": periodic geodesics that loop around just once.
The "covering space" philosophy described in example A) can now be taken literally! If the Riemannian manifold M' is a covering space of M, any prime geodesic P in M defines a deck transformation of M'. This transformation acts on the set X of prime geodesics sitting over P, so we get a onetoone and onto map
f: X → X
This is exactly like the "Frobenius automorphism" in number theory!
All this is particularly interesting when our manifold is a quotient of the upper halfplane by a discrete group  see "week215" for more on this. The reason is that some of these quotients are related to number theory. So, we get some direct interactions with example A).
D) The zeta function of a graph. We can take the idea of "periodic geodesic on a Riemannian manifold" and vastly simplify it by looking at closed loops in a graph with finitely many edges and vertices. We get a ζ function
ζ(s) = ∏ 1/(1  P^{s})
where P ranges over all "prime loops" in our graph: loops that don't backtrack or loop around more than once. Now P is the exponential of the length of the loop.
The "covering space" philosophy still applies, since we can define what it means for a graph G' to be a covering space of a graph G. Any prime loop P in G defines a deck transformation of G'. This acts on the set X of prime loops sitting over P, so we get a onetoone and onto map
f: X → X
which again deserves to be called the "Frobenius automorphism".
E) The zeta function of an affine scheme. Given a commutative ring, we can think of it as the ring of functions on some space. The ζ function of the commutative ring then counts the points of this space.
To make this precise, we cleverly invent a kind of space called an "affine scheme", which is secretly just another name for a commutative ring! So, any commutative ring R gives an affine scheme called Spec(R), purely by our fiendish definition.
If we take a function and evaluate it at a point, we should get a number. This should give a homomorphism from functions to numbers. But in algebraic geometry, "numbers" can be elements of any field k. So, let's say the "kpoints" of Spec(R) are the homomorphisms from R to k.
(This is a bit nontraditional, but I really need this here. For a more traditional alternative, see "week205".)
In particular, for any prime p we can take k to be the algebraic closure of the field with p elements. Let X be the set of kpoints of some affine scheme Spec(R). Then comes something wonderful: if x is a kpoint, so is x^{p}, since "raising to the pth power" is an automorphism of k. So, we get a map
f: X → X
sending x to x^{p}. This is called the "Frobenius automorphism"!
Since f is a discrete dynamical system, we can define its zeta function as in example B):
ζ(s) = ∏ 1/(1  P^{s})
where P ranges over all periodic orbits, and P is the exponential of the size of the periodic orbit, defined using the base p.
So far, this is the zeta function of our affine scheme "localized at p". If we multiply a bunch of factors like this, one for each ordinary prime p, we get the zeta function of our affine scheme. For example, the affine scheme Spec(Z) gives the Riemann zeta function.
In fact, all of example A) fits neatly into this one as a subexample. And if you know about schemes that aren't affine  like projective varieties, such as elliptic curves and other curves  you'll see this definition works for them too.
If you know someone else's definition of the zeta function of a scheme, it may not look like what I wrote here! But don't panic. The reason is that people like to express the zeta function of a discrete dynamical system f: X → X in terms of the number of fixed points of f^{ n}. When f is the Frobenius automorphism, these are usually called "points defined over the field with p^{n} elements". So, you'll see lots of formulas for zeta functions phrased in terms of these....
Okay. Enough examples.
There are a lot more, but I think these are the simplest. I hope you see how all these examples are just different expressions of the same idea. To go further, I would tell you how there are versions of the Riemann Hypothesis in all these examples, and also things called "Lfunctions", and lots of theorems and conjectures concerning them, too!
It's a wonderful example of the unity of mathematics. But, it's also a wonderful example of how this unity is obscured by people who zoom in on their own favorite special case and its particular technical complexities while never discussing the big picture. You wouldn't believe how hard it's been for me to figure out what I just told you! It's like trying to learn English by reading the US legal code, or learning basic chord progressions by listening to Schoenberg.
If you're just trying to get started, here's one of the more readable introductions:
5) Anton Deitman, Panorama of zeta functions, available as math.NT/0210060.
Audrey Terras has a lot of nice slide presentations about the ζ functions and Lfunctions of graphs:
6) Audrey Terras, Artin Lfunctions of graph coverings, available at http://math.ucsd.edu/~aterras/artin1.pdf
More on Lfunctions, available at http://math.ucsd.edu/~aterras/artin2.pdf
Here's a paper written in broken English but making a very serious attempt to explain things to the nonexpert:
7) David Zywina, The zeta function of a graph, available at http://math.berkeley.edu/~zywina/zeta.pdf
He gives a characterization of graphs whose zeta functions satisfy an analogue of the Riemann Hypothesis. Strangely, this analogue involves poles of the ζ function in the critical strip
0 < Re(s) < 1
Is this a real difference or the result of some difference in conventions?
Finally, I should explain some more technical stuff about ζ functions and fixed points, just so I don't forget it. Suppose we have a discrete dynamical system
f: X → X
and let
fix(f^{ n})
be the number of fixed points of the nth iterate of f.
We can define a weird function like this:
Z(u) = exp(∑_{ n>0} fix(f^{ n}) u^{n} / n)
You'll often see formulas like this running around, especially when f is some sort of "Frobenius automorphism". Sometimes people even call these guys zeta functions. But what in the world do they have to do with zeta functions???
Let's see. Suppose first that X consists of a single cycle of length k. Then f^{ n} has k fixed points when n is a multiple of k, but none otherwise, so:
Z(u) = exp( ku^{k}/k + ku^{2k}/2k + ku^{3k}/3k + ... ) = exp( u^{k} + u^{2k}/2 + u^{3k}/3 + ... ) = exp(ln(1/[1  u^{k})]) = 1/(1  u^{k})This is starting to look more like the zeta functions we know and love. It looks even better if we pick some prime number p and define
ζ(s) = Z(p^{s})
Then we get
ζ(s) = 1/(1  p^{ks})
which is precisely what we'd get using the definition in example B).
Furthermore, for both that old definition and our new one, the zeta function of a disjoint union of discrete dynamical systems is the product of the zeta functions of the parts. Since every discrete dynamical system is a disjoint union of cycles, we conclude that the definitions always agree. In other words,
ζ(s) = Z(p^{s})
with
Z(u) = exp(∑_{n>0} fix(f^{ n}) u^{n} / n)
is always equal to the zeta function defined in example B).
So, don't let anyone fool you  there aren't lots of completely different kinds of zeta functions! There's just a few kinds, and we could probably boil them down to just ONE kind with some work.
© 2005 John Baez
baez@math.removethis.ucr.andthis.edu
