Recently, my friend Yisong blogged about The Law of Quadratic Reciprocity, a proof of which made up my favorite chapter in An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery. The result isn’t particularly inspiring (although it is rather miraculous), but the proof Niven gives is so elegant and wonderful that it has been my favorite mathematical proof ever since I first encountered it.
The Law of Quadratic Reciprocity is not, however, my favorite mathematical result. That rather dubious honor falls to Hardy and Ramanujan’s formula for the Partition Function. Given a positive integer n, we define P(n) to be the number of unique ways (up to commutativity) of writing n as a sum of positive integers. This is such a simple concept that an elementary school student can reliably be called upon to calculate the first few values of the function (P(1)=1 [1], P(2)=2 [2, 1+1], P(3)=3 [3, 2+1, 1+1+1], P(4)=5 [4, 3+1, 2+2, 2+1+1, 1+1+1+1] and so on) until he or she gets bored and decides to go do something more immediately gratifying, probably involving television.
I first formally encountered the Partition Function in an advanced undergraduate class on set theory and topology taught by one of my favorite professors at Illinois, Paul Schupp. I had been studying recurrence relations at the time in an effort to explain them to my then-girlfriend, and once I found that actual mathematicians had deigned to give this ridiculously simple concept a name, I figured it would be a good exercise to work out the closed-form solution. I didn’t anticipate much trouble since the base cases were trivial and it was obvious that P(n) depended on P(n-1), and so on.
After a few days of absolute frustration, I gave up and asked Professor Schupp for the answer. I must have caught him at a bad time, because he just frowned and said he didn’t know of a closed-form expression. Unwilling to believe that the problem had never been polished off by some enterprising graduate student during his or her lunch hour, I ventured off to the dusty recesses of the math library and came away several hours later utterly amazed.
In fact, the Partition Function was first studied extensively by Euler, who gave the following generating function for P(n):
This formula is nice, but not particularly useful if one just wants the value of P(367) and doesn’t care about P(1) through P(366). In fact, my earlier inclination about the connection to recurrences was not totally off the wall, since P(n) does satisfy an elegant relation over the generalized pentagonal numbers:
However, it’s the convergent series expression for P(n) that’s truly amazing:
The above formula is actually due to Rademacher in 1937, but was largely the result of work done by Hardy and Ramanujan in 1918 (they proposed an asymptotic bound equivalent to the first term in the summation). The proof of this theorem and the story behind its development are both beyond the purview of this blog as well as the comprehension of this blogger, but before you pass on, dear reader, cast thine eye again on yonder equation. Consider a concept so simple that even a child with little formal education can grasp it, yet so vastly complex that it took two of the greatest mathematical minds in the history of human existence to fully describe. People who claim not to understand how math can be beautiful, I believe, have simply never been introduced to the Partition Function.
No matter what success my professional career is met with in the years to come, I will always profoundly regret that I posses neither the intellect nor the background to be a research mathematician.
