Monday, September 12, 2005

Of Partitions and Primes

I must have been hiding under a rock or something. I've only just found out about a big mathematical result about the partition function, p(n). p(n) is the number of ways of writing n as distinct (unordered) sums of integers. p(4) is 5 as 4 = 4+1 = 2+2 = 2+1+1 = 1+1+1+1.

First a digression: There's a bizarre formula for p(n) due to Rademacher and you can see it here. Note the n-1/24 subexpression. You might wonder where that 1/24 comes comes from. Actually it has a neat physical interpretation. In bosonic string theory you have a string which can oscillate with different harmonics. Given a string with a certain amount of energy there are different ways the energy could be shared out between the different harmonics. In quantum mechanics the amount of energy that can go into each harmonic is 'quantised' so that it is proportional to an integer. In addition the energy of a single quantum of energy in the nth harmomic is proportional to n. What this means is that if you have m1 quanta in the lowest harmonic, m2 in the second and so on then the total energy in the system, in suitable units, is E=m1+2*m2+3*m3+... In other words the number of different states with energy E is just the partition function p(E). In practice it's a little more complex because strings oscillate in a high dimensional background space so in fact there are harmonics in different directions. But essentially it's a similar counting problem. What I've actually described is a type of conformal field theory. Bosonic string theory (which is a 26D theory) is basically 24 copies of this system.

The interesting twist with quantum string theory is that when you consider the minimum energy you can put into each harmonic it isn't zero. You can argue this from the Hesenberg uncertainty principle, it can't be at rest because then you'd know its position and momentum simultaneously. So it must have non-zero energy. You can treat each harmonic independently so each harmonic has some energy. In fact they each have a minimum of half a quantum. (This is standard stuff.) So the correct formula is really E=(m1+1/2)+2*(m2+1/2)+3*(m3+1/2)+... The catch is that when you try to add up you get a divergent series. But this is no problem for physicists as you can sum the series using the methods outlined here. In fact, the 'zero-point' energy is essentially 1/2(1+2+3+...) so zeta regularisation gives -1/24. So the partition function of n is actually a function of the energy of a conformal field theory in its nth excited state. As I've obsessed about before, this is the same 24 that is the dimensionality of the Leech lattice, Golay code and lots of other good stuff. It's all connected mysteriously! Somehow this information about the quantum theory of strings is already encoded in the properties of the partition function discovered 70 years ago even though the partition function is defined by the elementary properties of numbers. This is also why I think String Theory is cool even if it might be useless as a physical theory. Even if we wanted to give up the string theory 'research programme' today it'd still be there hidden away inside the mathematics we study, regardless of whether we think that mathematics has physical applications.

I'll stop there as I don't even pretend to understand the connections. But do check out the proof of Rademacher's formula via Farey sequences and Ford circles. It's in a few analytic number theory books (eg. Apostol's Springer text on the subject) and it's one of the more beautiful proofs out there.

Anyway, that's all irrelevant. What's relevant is that the partition function has neat congruence properties. Ramanujan proved p(5m+4) is a multiple of 5 for all natural numbers m. He also showed p(7m+5) is a multiple of 7 and p(11m+6) is a multiple of 11. Many years ago Dyson explained the 5 and 7 cases using what was called the "crank conjecture" and then later this was extended to 11. But in the last couple of months Mahlburg has proved that the crank conjecture applies for all primes giving a way to produce such congruences for any prime. This is a pretty major result. People have been investigating these congruences at least since Ramanujan's day but now we have a handle on all of them simultaneously. More detail is posted here and this brings to the end a long chapter in number theory. Note, by the way, how often the number 24 appears through out this paper. :-)

Update: I've edited this so it makes slightly more sense.

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