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 other 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.  

Understanding the hidden patterns in the prime numbers is one of enduring
legacies of the ancient Greeks, who were the first to give a 
mathematical proof that there are infinitely many of them.  And since 
that time, many new and wonderous properties have been discovered:
For example, based on some numerical evidence (tables of calculations), 
Gauss conjectured in 1792, at the age of 15, that there are roughly
x/ln(x) (here, ln(x) denotes the logarithm to the base e of x) 
of them in the interval [1,x]; so, for example, for 

x = 1 billion = 10^9, 

one expects that there are about 

(10^9)/(ln(10^9)) = 48.25... million

of them less than 1 billion.  Then, some decades later, in the middle of 
the 19th century the brilliant polymath Riemann worked out
a beautiful formula which linked much more precise distribution questions
about the primes to the arrangement of the ``zeros of the Riemann zeta 
function in the critical strip''.  Among the myriad consequences of 
Riemann's profound insights was that Gauss's conjecture is equivalent
to a certain problem in the ``theory of analytic functions'', 
seemingly far-removed from prime numbers.  Despite finding this 
connection, 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 de la Vallee Poussin.  

One deep and mysterious problem remains from this grand programme to 
understand the distribution of prime numbers:  The Riemann Hypothesis.
To understand it better, in the work leading
up to his talk at IAS, Montgomery wondered what sort of ramifications 
there would be if it were true; specifically, assuming RH, 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 digits.)

What pattern do these numbers have?  Well, first of all, they get closer
and closer together the further along the list one goes, so it is 
natural to try to spread them out somehow by rescaling appropriately.
This Montgomery did, and what he expected to find was that 
the scaled zeros behaved more or less like a random sequence (specifically,
a ``Poisson process with lambda = 1'').  Instead, they appeared to behave
very strangely.  For one thing, they rarely seemed to clump together -- 
it was as though they ``repelled'' one another. 
Even stranger was that the consecutive pairs of these numbers
seemed rarely to get within a whole number distance of each other:  
They ``didn't like'' being a distance near to 1, 2, 3, 4, ... apart, 
but 1.1 or 2.1 seemed ok.

Suspecting that there might be even deeper patterns,  
Montgomery fit a smooth curve to the ``distribution function'' that
governed the differences between these consecutive normalized
zeta zeros, 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, when he presented his results, there was 
someone in the audience who had seen this distribution before:  Freeman
Dyson, the great British physicist and mathematician, and permanent 
member at the IAS.  

Dyson got his academic start as a student of Besicovich, though he also
dabbled in number theory, taking classes and working with the great 
Cambridge University number theorist G. H. Hardy.  Among his 
accomplishments in this field (number theory) are the concept of the
``crank of a partition'' and the ``Thue-Siegel-Dyson theorem''.  
Though his early interests were largely mathematical, at some point he 
switched almost totally to working on physics, and became one of the 
scientists who worked with J. Robert Oppenheimer; though, apparently, 
he was not directly involved in the Manhattan Project.   


After Montgomery's talk, Dyson came up to him and informed him that the 
distribution function he had stumbled upon is, in fact, the exact same that
comes up when studying the energy levels of atoms!  More precisely,
the distribution was the ``pair correlation function 
for eigenvalues of random matrices'', a common mathematical object that 
comes up in some parts of quantum mechanics.

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-studied topic of certain prominent mathematicians.  Physicists have also
added some of their intuition and insight, and there is a
feeling by some that perhaps there are even deeper connections 
between zeta functions (and prime numbers) and energy levels of atoms.