Partition function (number theory): Difference between revisions

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:<math> p(n) \sim \frac{\exp\left(\pi\sqrt{2/3}\sqrt n\right)}{4n\sqrt3} .</math>
:<math> p(n) \sim \frac{\exp\left(\pi\sqrt{2/3}\sqrt n\right)}{4n\sqrt3} .</math>
==References==
* {{cite book | author=Tom M. Apostol | title=Modular functions and Dirichlet Series in Number Theory  | edition=2nd ed | series=[[Graduate Texts in Mathematics]] | volume=41 | publisher=[[Springer-Verlag]] | year=1990 | isbn=0-387-97127-0 | pages=94-112 }}
* {{cite book | author=G.H. Hardy | authorlink=G. H. Hardy | coauthors=[[E. M. Wright]] | title=An Introduction to the Theory of Numbers | edition=6th ed. | publisher=[[Oxford University Press]] | year=2008 | isbn=0-19-921986-5 | pages=361-392 }}

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In number theory the partition function p(n) counts the number of partitions of a positive integer n, that is, the number of ways of expressing n as a sum of positive integers (where order is not significant).

Thus p(3) = 3, since the number 3 has 3 partitions:

  • 3
  • 2+1
  • 1+1+1

Properties

The partition function satisfies an asymptotic relation