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Hardy-Ramanujan’s asymptotic formula for partitions and the central limit theorem. (English) Zbl 0873.60010

Summary: 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 \(t\to 1\), the normalized \(X_t\) are asymptotically Gaussian. The mode of convergence is sufficiently strong for the conclusion of a local and central limit theorem to hold, leading to the classical formula of Hardy-Ramanujan, \(p(n)\sim \exp(\pi\sqrt {2/3}\sqrt n)/(4n\sqrt 3)\).

MSC:

60F05 Central limit and other weak theorems
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