Partition function (number theory)

In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. For example, 4 can be partitioned in 5 distinct ways

Rademacher's series

An asymptotic expression for p(n) is given by

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:p(n) sim rac {exp left( pi sqrt {2n/3} ight) } {4nsqrt{3}} mbox { as } n ightarrow infty

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This expression was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920.

Related Topics:
G. H. Hardy - Ramanujan - 1918 - J. V. Uspensky - 1920

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In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan's results by providing a convergent series expression for p(n). It is

Related Topics:
1937 - Hans Rademacher

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:p(n)=rac{1}{pi sqrt{2}} sum_{k=1}^infty A_k(n);

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sqrt{k} ; rac{d}{dn}

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left( rac {sinh left( rac{pi}{k}

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sqrt{rac{2}{3}left(n-rac{1}{24} ight)} ight) }

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where

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:A_k(n) = sum_{0le m < k ; ; ; (m,k)=1}exp left(

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pi i s(m,k) - 2pi inm/k ight).

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Here, the notation (m,n)=1 implies that the sum should occur only over the values of m that are relatively prime to n. The function s(m,k) is a Dedekind sum. The proof of Rademacher's formula is interesting in that it involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function in a central way.

Related Topics:
Relatively prime - Dedekind sum - Ford circle - Farey sequence - Modular symmetry - Dedekind eta function

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