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May 30, 2005

This Week's Finds in Mathematical Physics - Week 217

John Baez

Last week I described lots of different zeta functions, but didn't say much

about what they're good for. This week I'd like to get started on fixing

that problem.

People have made lots of big conjectures related to zeta functions.

So far they've just proved just a few... but it's still a big deal.

For example, Andrew Wiles' proof of Fermat's Last Theorem was just

a tiny spin-off of his work on something much bigger called the

Taniyama-Shimura conjecture. Now, personally, I think Fermat's Last

Theorem is a ridiculous thing. The last thing I'd ever want to know is

whether this equation:

x^n + y^n = z^n

has nontrivial integer solutions for n > 2. But the Taniyama-Shimura

Conjecture is really interesting! It's all about the connection between

geometry, complex analysis and arithmetic, and it ties together some big

ideas in an unexpected way. This is how it usually works in number theory:

cute but goofy puzzles get solved as a side-effect of deep and interesting

results related to zeta functions and L-functions - sort of like how the

powdered drink "Tang" was invented as a spinoff of going to the moon.

For a good popular book on Fermat's Last Theorem and the Taniyama-Shimura

Conjecture, try:

1) Simon Singh, Fermat's Enigma: The Epic Quest to Solve the World's

Greatest Mathematical Problem, Walker, New York, 1997.

Despite the "world's greatest mathematical problem" baloney, this book does

a great job of telling the story without drowning the reader in math.

But you read This Week's Finds because you *want* to be drowned in math,

and I wouldn't want to disappoint you. So, let me list a few of the big

conjectures and theorems related to zeta functions.

Here goes:

A) The Riemann Hypothesis - the zeros of the Riemann zeta function in

the critical strip

0 <= Re(s) <= 1

actually lie on the line Re(s) = 1/2.

First stated in 1859 by Bernhard Riemann; still open.

This implies a good estimate on the number of primes less than a given

number, as described in "week216".

B) The Generalized Riemann Hypothesis - the zeros of any Dirichlet L-function

that lie in the critical strip actually lie on the line Re(s) = 1/2.

Still open, since the Riemann Hypothesis is a special case.

A "Dirichlet L-function" is a function like this:

L(chi,s) = sum_{n > 0} chi(n)/n^s

where chi is any "Dirichlet character", meaning a periodic complex function

on the positive integers such that

chi(nm) = chi(n) chi(m)

If we take chi = 1 we get back the Riemann zeta function.

Dirichlet used these L-functions to prove that there are infinitely many

primes equal to k mod n as long as k is relatively prime to n. The

Generalized Riemann Hypothesis would give a good estimate on the number

of such primes less than a given number, just as the Riemann Hypothesis

does for plain old primes.

Erich Hecke established the basic properties of Dirichlet L-functions

in 1936, including a special symmetry called the "functional equation"

which Riemann had already shown for his zeta function. So I bet Hecke

must have dreamt of the Generalized Riemann Hypothesis, even if he didn't

dare state it.

C) The Extended Riemann Hypothesis - for any number field, the zeros of its

zeta function in the critical strip actually lie on the line Re(s) = 1/2.

Still open, since the Riemann Hypothesis is a special case.

I described the zeta functions of number fields in "week216".

These are usually called "Dedekind zeta functions". Hecke also

proved a functional equation for these back in 1936.

D) The Grand Riemann Hypothesis - for any automorphic L-function,

the zeros in the critical strip actually lie on the line Re(s) = 1/2.

This is still open too, since it includes A)-C) as special cases!

I don't want to tell you what "automorphic L-functions" are yet.

For now, you can just think of them as grand generalizations of both

Dirichlet L-functions and zeta functions of number fields.

E) The Weil Conjectures - The zeros of the zeta function of any smooth

algebraic variety over a finite field lie on the line Re(s) = 1/2.

Also: such zeta functions are quotients of polynomials, they satisfy a

functional equation, and they can be computed in terms of the topology

of the corresponding *complex* algebraic varieties.

First stated in 1949 by Andre Weil; proof completed by Pierre Deligne

in 1974 based on much work by Michael Artin, J.-L. Verdier, and especially

Alexander Grothendieck. Grothendieck invented topos theory as part of

the attack on this problem!

F) The Taniyama-Shimura Conjecture - every elliptic curve over the rational

numbers is a modular curve. Or, equivalently: every L-function of an

elliptic curve is the L-function of a modular curve.

This was first conjectured in 1955 by Yukata Taniyama, who worked on it

with Goto Shimura until committing suicide in 1958. Around 1982 Gerhard

Frey suggested that this conjecture would imply Fermat's Last Theorem; this

was proved in 1986 by Ken Ribet. In 1995 Andrew Wiles and Richard Taylor

proved a big enough special case of the Taniyama-Shimura Conjecture to get

Fermat's Last Theorem. The full conjecture was shown in 1999 by Breuil,

Conrad, Diamond, and Taylor.

I don't want to say what L-functions of curves are yet... but they are

a lot like Dirichlet L-functions.

G) The Langlands Conjectures - Any automorphic representation pi of a

connected reductive group G, together with a finite-dimensional representation

of its L-group, gives an automorphic L-function L(s,pi). Also: these

L-functions all satisfy functional equations. Furthermore, they depend

functorially on the group G, its L-group, and their representations.

Zounds! Don't worry if this sounds like complete gobbledygook! I only

mention it to show how scary math can get. As Stephen Gelbart once wrote:

The conjectures of Langlands just alluded to amount (roughly)

to the assertion that the other zeta-functions arising in

number theory are but special realizations of these L(s,pi).

Herein lies the agony as well as the ecstacy of Langlands'

program. To merely state the conjectures correctly requires

much of the machinery of class field theory, the structure

theory of algebraic groups, the representation theory of real

and p-adic groups, and (at least) the language of algebraic

geometry. In other words, though the promised rewards are

great, the initiation process is forbidding.

I hope someday I'll understand this stuff well enough to say something more

helpful! Lately I've been catching little glimpses of what it's about....

But, right now I think it's best if I talk about the "functional equation"

satisfied by the Riemann zeta function, since this gives the quickest way

to see some of the strange things that are going on.

The Riemann zeta function starts out life as a sum:

zeta(s) = 1^{-s} + 2^{-s} + 3^{-s} + 4^{-s} + ....

This only converges for Re(s) > 1. It blows up as we approach s = 1,

since then we get the series

1/1 + 1/2 + 1/3 + 1/4 + ....

which diverges. However, in 1859 Riemann showed that we can analytically

continue the zeta function to the whole complex plane except for this pole

at s = 1.

He also showed that the zeta function has an unexpected symmetry:

its value at any complex number s is closely related to its value at 1-s.

It's not true that zeta(s) = zeta(1-s), but something similar is true,

where we multiply the zeta function by an extra fudge factor.

To be precise: if we form the function

pi^{-s/2} Gamma(s/2) zeta(s)

then this function is unchanged by the transformation

s |-> 1 - s

This symmetry maps the line

Re(s) = 1/2

to itself, and the Riemann Hypothesis says all the zeta zeros in

the critical strip actually lie on this magic line.

This symmetry is called the "functional equation". It's the tiny tip of a

peninsula of a vast and mysterious continent which mathematicians are still

struggling to explore. Riemann gave two proofs of this equation. You can

find a precise statement and a version of Riemann's second proof here:

2) Daniel Bump, Zeta Function, lecture notes on "the functional

equation" available at http://math.stanford.edu/~bump/zeta.html

and http://www.maths.ex.ac.uk/~mwatkins/zeta/fnleqn.htm

This proof is a beautiful application of Fourier analysis. Everyone

should learn it, but let me try to sketch the essential idea here.

I will deliberately be VERY rough, and use some simplified nonstandard

definitions, since the precise details have a way of distracting your

eye just as the magician pulls the rabbit out of the hat.

We start with the function zeta(2s):

1^{-s} + 4^{-s} + 9^{-s} + 25^{-s} + ....

Then we apply a curious thing called the "Mellin transform", which turns

this function into

z^{1} + z^{4} + z^{9} + z^{25} + ....

Weird, huh? This is almost the "theta function"

theta(t) = sum_n exp(pi i n^2 t)

where we sum over all integers n. Indeed, it's easy to see that

(theta(t) - 1)/2 = z^{1} + z^{4} + z^{9} + z^{25} + ....

when

z = exp(pi i t)

The theta function transforms in a very simple way when we replace

t by -1/t, as one can show using Fourier analysis.

Unravelling the consequences, this implies that the zeta function

transforms in a simple way when we replace s by 1-s. You have to

go through the calculation to see precisely how this works... but

the basic idea is: a symmetry in the theta function yields a symmetry

in the zeta function.

Hmm, I'm not sure that explained anything! But I hope at least the

mystery is more evident now. A bunch of weird tricks, and then *presto* -

the functional equation! To make progress on understanding the Riemann

Hypothesis and its descendants, we need to get what's going on here.

I feel I *do* get the Mellin transform; I'll say more about that later.

But for now, note that the theta function transforms in a simple way, not

just when we do this:

t |-> -1/t

but also when we do this:

t |-> t + 2

Indeed, it doesn't change at all when we add 2 to t, since exp(2 pi i) = 1.

Now, the maps

t |-> -1/t

and

t |-> t + 1

generate the group of all maps

at + b

t |-> --------

ct + d

where a,b,c,d form a 2x2 matrix of integers with determinant 1.

These maps form a group called PSL(2,Z), or the "modular group".

A function that transforms simply under this group and doesn't blow up

in nasty ways is called a "modular form". In "week197" I gave the precise

definition of what counts as transforming simply and not blowing up in

nasty ways. I also explained how modular forms are related to elliptic

curves and string theory. So, please either reread "week197" or take my

word for it: modular forms are cool!

The theta function is almost a modular form, but not quite. It doesn't

blow up in nasty ways. However, it only transforms simply under a subgroup

of PSL(2,Z), namely that generated by

t |-> -1/t

and

t |-> t + 2

So, the theta function isn't a full-fledged modular form.

But since it comes close, we call it an "automorphic form".

Indeed, for any discrete subgroup G of PSL(2,Z), functions that transform

nicely under G and don't blow up in nasty ways are called "automorphic forms"

for G. They act a lot like modular forms, and people know vast amounts

about them. It's the power of automorphic forms that makes number theory

what it is today!

We can summarize everything so far in this slogan:

THE FUNCTIONAL EQUATION FOR THE RIEMANN ZETA FUNCTION SAYS

"THE THETA FUNCTION IS AN AUTOMORPHIC FORM"

Before you start printing out bumper stickers, I should explain....

The point of this slogan is this. We *thought* we were interested in

the Riemann zeta function for its own sake, or what it could tell us

about prime numbers. But with the wisdom of hindsight, the first thing we

should do is hit this function with the Mellin transform and repackage all

its information into an automorphic form - the theta function.

Zeta is dead, long live theta!

The Riemann zeta function is just like all the fancier zeta functions and

L-functions in this respect. The fact that they satisfy a "functional

equation" is just another way of saying their Mellin transforms are

automorphic forms... and it's these automorphic forms that exhibit the

deeper aspects of what's going on.

Now let me say a little bit about the Mellin transform.

Ignoring various fudge factors, the Mellin transform is basically just

the linear map that sends any function of s like this:

n^{-s}

to this function of z:

z^n

In other words, it basically just turns things upside down, replacing the

base by the exponent and vice versa. The minus sign is just a matter of

convention; don't worry about that too much.

So, the Mellin transform basically sends any function like this, called a

"Dirichlet series":

a_1 1^{-s} + a_2 2^{-s} + a_3 3^{-s} + a_4 4^{-s} + ....

to this function, called a "Taylor series":

a_1 z^1 + a_2 z^2 + a_3 z^3 + a_4 z^4 + ....

Now, why would we want to do this?

The reason is that multiplying Taylor series is closely related to *addition*

of natural numbers:

z^n z^m = z^{n+m}

while multiplying Dirichlet series is closely related to *multiplication*

of natural numbers:

n^{-s} m^{-s} = (nm)^{-s}

The Mellin transform (and its inverse) are how we switch between these two

pleasant setups!

Indeed, it's all about algebra - at least at first. We can add natural

numbers and multiply them, so N becomes a monoid in two ways. A "monoid",

recall, is a set with a binary associative product and unit. So, we have

two closely related monoids:

(N,+,0)

and

(N,x,1)

Given a monoid, we can form something called its "monoid algebra" by taking

formal complex linear combinations of monoid elements. We multiply these

in the obvious way, using the product in our monoid.

If we take the monoid algebra of (N,+,0), we get the algebra of Taylor

series! If we take the monoid algebra of (N,x,1), we get the algebra of

Dirichlet series!

(Actually, this is only true if we allow ourselves to use *infinite* linear

combinations of monoid elements in our monoid algebra. So, let's do that.

If we used finite linear combinations, as people often do, (N,+,0) would give

us the algebra of polynomials, while (N,x,0) would give us the algebra of

"Dirichlet polynomials".)

Of course, algebraically we can combine these structures. (N,+,x,0,1) is

a rig, and by taking formal complex linear combinations of natural numbers

we get a "rig algebra" with two products: the usual product of Taylor series,

and the usual product of Dirichlet series. They're compatible, too, since

one distributes over the other. They both distribute over addition.

However, if we're trying to get an algebra of functions on the complex plane,

with pointwise multiplication as the product, we need to make up our mind:

either Taylor series or Dirichlet series! We then need the Mellin transform

to translate between the two.

So, what seems to be going on is that people take a puzzle, like

"what is the sum of the squares of the divisors of n?"

or

"how many ideals of order n are there in this number field?"

and they call the answer a_n.

Then they encode this sequence as either a Dirichlet series:

a_1 1^{-s} + a_2 2^{-s} + a_3 3^{-s} + a_4 4^{-s} + ....

or a Taylor series:

a_1 z^1 + a_2 z^2 + a_3 z^3 + a_4 z^4 + ....

The first format is nice because it gets along well with multiplication of

natural numbers. For example, in our puzzle about ideals, every ideal is

a product of prime ideals, and its norm is the product of the norms of those

prime ideals... so our Dirichlet series will have an Euler product formula.

The second format is nice *if* our Taylor series is an automorphic form.

This will happen precisely when our Dirichlet series satisfies a functional

equation.

(For experts: I'm ignoring some fudge factors involving the gamma function.)

I still need to say more about *which* puzzles give automorphic forms,

what it really means when they *do*. But, not this week! I'm tired,

and I bet you are too.

For now, let me just give some references. There's a vast amount of material

on all these subjects, and I've already referred to lots of it. But right now

I want to focus on stuff that's free online, especially stuff that's readable

by anyone with a solid math background - not journal articles for experts, but

not fluff, either.

Here's some information on the Riemann Hypothesis provided by the Clay

Mathematics Institute, which is offering a million dollars for its solution:

3) Clay Mathematics Institute, Problems of the Millenium:

the Riemann Hypothesis, http://www.claymath.org/millennium/ [Broken]

The official problem description by Enrico Bombieri talks about evidence

for the Riemann Hypothesis, including the Weil Conjectures. The article by

Peter Sarnak describes generalizations leading up to the Grand Riemann

Hypothesis. In particular, he gives a super-rapid introduction to

automorphic L-functions.

Here's a nice webpage that sketches Wiles and Taylor's proof of Fermat's last

theorem:

4) Charles Daney, The Mathematics of Fermat's Last Theorem,

http://www.mbay.net/~cgd/flt/fltmain.htm [Broken]

I like the quick introductions to "Elliptic curves and elliptic functions",

"Elliptic curves and modular functions", "Zeta and L-functions", and "Galois

Representations" - they're neither too detailed nor too vague, at least for

me.

Here's a nice little intro to the Weil Conjectures:

5) Runar Ile, Introduction to the Weil Conjectures,

http://folk.uio.no/~ile/WeilA4.pdf [Broken]

James Milne goes a lot deeper - his course notes on etale cohomology include

a proof of the Weil Conjectures:

6) James Milne, Lectures on Etale Cohomology,

http://www.jmilne.org/math/CourseNotes/math732.html [Broken]

while his course notes on elliptic curves sketch the proof of Fermat's Last

Theorem:

7) James Milne, Elliptic Curves,

http://www.jmilne.org/math/CourseNotes/math679.html [Broken]

Here's a nice history of what I've been calling the Taniyama-Shimura

Conjecture, which explains why some people call it the Taniyama-Shimura-Weil

conjecture, or other things:

8) Serge Lang, Some history of the Shimura-Taniyama Conjecture,

AMS Notices 42 (November 1995), 1301-1307. Available at

http://www.ams.org/notices/199511/forum.pdf

Here's a quick introduction to the proof of this conjecture, whatever

it's called:

9) Henri Diamond, A proof of the full Shimura-Taniyama-Weil Conjecture

is announced, AMS Notices 46 (December 1999), 1397-1401. Available

at http://www.ams.org/notices/199911/comm-darmon.pdf

I won't give any references to the Langlands Conjectures, since

I hope to talk a lot more about those some other time.

And, I hope to keep on understanding this stuff better and better!

Quote of the week:

"If I were to awaken after having slept for a thousand years, my

first question would be: Has the Riemann hypothesis been proven?" -

David Hilbert

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# This Week's Finds in Mathematical Physics (Week 217)

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