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Hardy–Ramanujan's Asymptotic Formula for Partitions and the Central Limit Theorem

✍ Scribed by Luis Báez-Duarte

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
668 KB
Volume
125
Category
Article
ISSN
0001-8708

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✦ Synopsis

Let f (z) be the generating function of the sequence [ p(n)] of unrestricted partitions of n, and let X t be an integral random variable taking the value n with probability ( f (t)) &1 p(n) t n . It is shown here that, as t Ä 1, the normalized X t are asymptotically Gaussian. The mode of convergence is sufficiently strong for the conclusion of a local central limit theorem to hold, leading to the classical formula of Hardy Ramanujan, p(n)texp(? -2Â3 -n)Â(4n -3). d dt m(t), (1.4) article no. AI971599 114

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