Recently there was an intriguing breakthrough in the study of partition numbers. partitions numbers P(n) count the number of ways of expressing n as a sum of positive integers. E.g. P(4) = 5 because 4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4. The sequence of partition numbers goes 1,1,2,3,5,7,11,15,22, … It starts off wanting to be the Fibonacci numbers but then after the first five it changes its mind and gives the first five prime numbers before degenerating into a rapidly increasing sequence of interesting numbers.
There are lots of well known things that can be said about the partition numbers but I’ll leave that to Ken Ono who is the main figure in the new discovery, actually two discoveries. The first discovery is a new finite formula for partition functions and the second is a new explanation for some congruence relations discovered by Rananujan. I highly recommend this low level lecture by Ken Ono as an introduction to the new finds.
One thing that is not mentioned in all the recent news coverage is the important connection between partition numbers and string theory that is very easy to see even a very basic level. From the theory of musical harmonics you know that a string has vibration modes labelled by integers k, whose frequency is ωk = kα for some constant α that depends on things like the tension in the string. When a string is quantized it can be treated like a set of decoupled harmonic oscillators with energy levels Ek = (1/2 + mk)ħωk where the sequnece of non-negative integers mk labels the eignestates of the oscillators. So the total energy is given by
E = Σk (1/2 + mk)ħkα
The zero-point energy is E0 = 1/2 ħkα(1+2+3+4+…) . We can either ignore this as an irrelevant constant while pretending not to notice that it is infinite, or we can use zeta regulation to deduce that 1+2+3+4+ … = ζ(-1) = -1/12. In any case, what we are really interested in is the rest of the sum and to understand it we just need a simple trick. Write mk k = ( k+k+…+k ) (mk times) Then you will immediately notice that the number of states with an energy En = E0 + nħα is exactly P(n), the partition number of n. This is also valid for the relativistic bosonic string in 26 dimensional spacetime, except that then you need to multiply by 24 because the one dimensional string can vibrate independently in any of the 24 space dimensions transverse to the string.
The partition function for bosonic string theory is therefore given by
Z = Σn P(n) exp( – (24n-1)ħα)
Perhaps that’s why they call it the partition function 🙂