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II. The Partition Function
18
Now we turn to the analytic number theory derivation of this
asymptotic formula.
The Generating Function

To put into sharp focus the fact that order does not count, we may
view p(n) as the number of representations of n as a sum of 1’s and
2’s and 3’s ..., etc. But this is just the “change making” problem
where coins come in all denominations. The analysis in that problem
extends verbatim to this one, even though we now have an infinite
number of coins, So we obtain

∞ ∞
1
n
p(n)z (1)
1 − z k
n 0 k 1
valid for |z| < 1, where we understand that p(0) 1.
Having thus obtained the generating function, we turn to the sec-
ond stage of attack, investigating the function. This is always the
tricky (creative?) part of the process. We know pretty well what kind
of information we desire about p(n): an estimate of its growth, per-
haps even an asymptotic formula if we are lucky. But we don’t know
exactly how this translates to the generating function. To grasp the
connection between the generating function and its coefficients, then,
seems to be the paramount step. How does one go from one to the
other? Mainly how does one go from a function to its coefficients?
It is here that complex numbers really play their most important
role. The point is that there are formulas (for said coefficients). Thus
n f (n) (0)
we learned in calculus that, if fàz) a n z , then a n ,
n!
expressing the desired coefficients in terms of high derivatives of the
function. But this a terrible way of getting at the thing. Except for
rare “made up” examples there is very little hope of obtaining the nth
derivative of a given function and even estimating these derivatives
is not a task with very good prospects. Face it, the calculus approach
is a flop.
Cauchy’s theorem gives a different and more promising approach.
n
Thus, again with fàz) a n z , this time we have the formula