nLab partition function (number theory)

This entry is about the notion in number theory/combinatorics. For partition functions in the sense of statistical mechanics and quantum field theory see at partition function.

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Idea

In number theory/combinatorics, the function which assigns to a natural number $n \in \mathbb{N}$ the number $p(n)$ of its partitions (into multisets of positive natural numbers) is often called the paritition function.

Properties

Generating function

Its (ordinary) generating function is

$$\sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty (1 - x^k)^{-1} \,.$$

Asymptotic expansion

An asymptotic expansion of the partition function is (Hardy & Ramanujan 1918, see DeSalvo 20 for discussion)

$$p(n) \;\sim\; \frac { e^{ \sqrt{2/3}\pi \cdot \sqrt{n} } } { 4 \sqrt{3} \cdot n} \,.$$

References

• On-Line Encyclopedia of Integer Sequences, OEIS A000041

See also:

• Wikipedia, Partition function (number theory)

Asymptotic behaviour:

• Godfrey Hardy, Srinivasa Ramanujan, Asymptotic formulaæ in combinatory analysis, Proceedings of the London Mathematical Society, 2(1):75–115, 1918 (doi:10.1112/plms/s2-17.1.75, pdf)

• Stephen DeSalvo, Will the real Hardy-Ramanujan formula please stand up? (arXiv:2003.06908)