Une \'etude asymptotique probabiliste des coefficients d'une s\'erie enti\`ere

TL;DR

This paper provides a probabilistic proof of the asymptotic formula for integer partitions, using convergence to Gaussian variables and establishing conditions for classical asymptotic results.

Contribution

It proves a Lyapunov-type theorem justifying Gaussian convergence and applies it to derive classical asymptotics for partitions with distinct parts.

Findings

Established a Lyapunov-type theorem for Gaussian convergence.

Validated the strong Gaussian type condition for classical partition asymptotics.

Derived asymptotic formulas for partitions with distinct parts.

Abstract

Following the ideas of Rosenbloom [7] and Hayman [5], Luis B\'aez-Duarte gives in [1] a probabilistic proof of Hardy-Ramanujan's asymptotic formula for the partitions of an integer. The main principle of the method relies on the convergence in law of a family of random variables to a gaussian variable. In our work we prove a theorem of the Liapounov type (Chung [2]) that justifies this convergence. To obtain simple asymptotic formul{\ae} a condition of the so-called strong Gaussian type defined by Luis B\'aez-Duarte is required; we demonstrate this in a situation that make it possible to obtain a classical asymptotic formula for the partitions of an integer with distinct parts (Erd\"os-Lehner [4], Ingham [6]).