Partition function (number theory)

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2020 Mathematics Subject Classification: Primary: 11P [MSN][ZBL]

A partition of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of 4 read: $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. The partition function $p(n)$ counts the number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [a1]) and Ramanujan discovered the surprising congruences $p(5m+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0 \pmod{11}$, and others. He also found the asymptotic relation $$ p(n) \sim \frac{e^{K \sqrt{n}}}{4n\sqrt{3}}\ \ \text{as}\ \ n \rightarrow \infty \ , $$ where $K = \pi\sqrt{2/3}$. Later this was completed to an exact series expansion by H. Rademacher (see [a2]).

One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [a3]). See also Additive number theory; Additive problems.


[a1] G.H. Hardy; E. M. Wright; An Introduction to the Theory of Numbers Oxford University Press (2008) ISBN 0-19-921986-5
[a2] Tom M. Apostol; Modular functions and Dirichlet Series in Number Theory Graduate Texts in Mathematics 41 Springer-Verlag (1990) ISBN 0-387-97127-0
[a3] G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976)
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