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In the following essay I have put technical concepts in double quotes,
to alert the reader. They can be safely ignored for the most part.
The account below is slightly fictionalized to make it accessible for
In 1972, during a visit to the Institute for Advanced Studies (IAS) near
Princeton University, Hugh Montgomery, a researcher in
number theory, presented some startling computations he had
made regarding prime numbers, which are whole numbers 2,3,5,7,...
that can only be factored into whole numbers in a trivial way.
The number 6 can be factored ``non-trivially'' as 6 = 2 x 3, so it
is not a prime number; a number like 11, however, can only be factored
as 11 = 1 x 11, so *is* a prime number.
The ancient Greeks first proved that there are infinitely many prime
numbers. Based on some numerical evidence (tables of calculations),
Gauss conjectured in 1792, at the age of 15, that there are roughly
x/ln(x) of them in the interval [1,x]; so, for example, for
x = 1 trillion = 10^12, one expects that there are about
(10^12)/ln(10^12) of them less than 1 trillion. Here, the `ln' denotes the
natural logarithm; that is, the logarithm base e = 2.71828...
In the middle of the 19th century, the brilliant polymath Riemann worked out
a beautiful formula, which is a bit too technical to describe in this
note, and which linked some fundamental questions about prime numbers to
the arrangement of the ``zeros of the Riemann zeta function in the critical
strip''. In particular, he showed that Gauss's conjecture is logically equivalent
to showing that the zeta function has ``no zeros on or near the 1 line''. Even so,
Riemann was not able to prove Guass's conjecture. That task was finally
completed just at the end of the 19th century by Hadamard and
independently by de la Vallee Poussin, and their result became known
as the Prime Number Theorem.
One nagging problem remains from this grand programme to understand the
distribution of prime numbers, and that is what is known as the
``Riemann Hypothesis''. It turns out that this conjecture about the
``zeros of the Riemann zeta function'' (which we won't bother to describe) is
*equivalent* to a certain very sharp estimate for the number of prime
numbers in the interval [1,x] -- sort of a refinement of the Prime Number
What Montgomery was working on at IAS is related to the Riemann Hypothesis,
and addressed the following basic question: suppose the Riemann Hypothesis
is true. What can one say about the ``zeros of the zeta function in the critical
strip''? Upon computing the ``absolute values of the imaginary components''
of these zeros, Montgomery was led to the list of numbers
14.1347, 21.0220, 25.0109, 30.4249, 32.9351, ...
(These are only approximations to 4 decimal places.)
What pattern do these numbers have? Well, first of all, they get closer
and closer together the further along the list one goes. A common trick
for spreading them out in a nice way, to make them easier to analyze,
is to multiply them by some ``scaling function to normalize the average
spacing between consecutive numbers to 1''.
Upon normalizing the spacing of the numbers, Montgomery expected to
find that they behaved more or less like a random sequence (specifically,
a ``Poisson process with lambda = 1''). Instead, they appeared to behave
very strangely: these normalized zeta zeros seemed to ``prefer'' not
being at certain distances from each other; for example, distances near to
1/2 turned out to be rarer than distances near to 3/4.
Montgomery fit a smooth curve to the ``distribution function'' that
governed the differences between pairs of normalized
zeta zeros in certain long intervals, and found that they seemed
to fit a quite peculiar function involving the constant pi and the sine
squared of the spacing under consideration. Peculiar indeed!
Fortunately for Montgomery, there was a physicist at IAS who had
seen this distribution before: Freeman Dyson.
Dsyon is a British physicist and mathematician, and is a permanent
member at IAS. He got his academic start working with the great
Cambridge University number theorist G. H. Hardy (though his
advisor was Besicovich). Indeed, Dyson
himself was once a researcher in number theory, and developed
a number of highly successful mathematical
concepts studied even today (like the ``crank of a partition'' and the
``Thue-Siegel-Dyson theorem''). At some point Dyson switched almost
totally to working on physics, and became one of the scientists who
worked with J. Robert Oppenheimer.
During teatime, Dyson informed Montgomery that the
distribution function he had stumbled upon is, in fact, the exact same that
comes up when studying the energy levels of atoms, in something called
the ``Gaussian Unitary Ensemble''! More precisely, the distribution Montgomery
discovered is the ``pair correlation function for eigenvalues of random matrices'',
a common mathematical object that comes up in some parts of quantum
Since this discovery of Montgomery, and the chance encounter with Dyson,
the subject of ``pair correlation of zeta zeros'' has exploded, and is a
well-researched topic of many prominent mathematicians. Physicists have also
added some of their intuition and insight to the subject, and there is a
feeling held by some that perhaps there are even deeper connections
between zeta functions (and prime numbers) and energy levels of atoms.
Edited by Ernie Croot on Mar 26, 2009 3:01 PM
Very interesting Ernie. Do you have any other resources (videos, links, illustrations) that would help people understand more about this?
Here is a link to an American Scientist article (which has a few pictures) I found while searching with Google:
Here is a more technical article written by a physicist (I haven't read it carefully, so can't judge how
good it is -- it also has pictures):
Here is another link I found just today (Monday, Jul 28), with some more recent work:
I remember seeing a particularly nice website some years ago, written by a physicist, that had some nice graphics and
videos. If I find it, I will post a link here.
Edited by Ernie Croot on Jul 28, 2008 9:27 AM