- #1
John Baez
Also available as http://math.ucr.edu/home/baez/week217.html
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/
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
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
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
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
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
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
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/
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
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
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
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
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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Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
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