Partition generation

Calculus Level 4

Let the partition function $P(n)$ enumerate the ways $n$ can be expressed as a distinct sum of positive integers, e.g. $P(4) = 5$ since $4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1$ are the only ways to represent $4$ .

$\prod_{p \ \text{prime}} \left[ \sum_{n=0}^{\infty} P(n)p^{-n} \right]$

Does the above product converge?

Yes No Undecidable in ZFC

1 solution

Jake Lai
Aug 17, 2015

It is known that $\displaystyle \sum_{n=0}^{\infty} P(n)x^{n} = \prod_{m=1}^{\infty} (1-x^{m})^{-1}$ . Converting the sum in question to a product, we get

$\prod_{p \ \text{prime}} \prod_{m=1}^{\infty} (1-p^{-m})^{-1} = \prod_{m=1}^{\infty} \zeta(m)$

Since $\displaystyle \lim_{s \to \infty} \zeta(s) = 1 \neq 0$ and $\zeta(1)$ is divergent, it must then be that the product diverges.

Argh... I thought it said "Does the above product diverge?"

Julian Poon - 5 years, 10 months ago

Partition generation

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