A Note on Goldbach Partitions of Large Even Integers

TL;DR

This paper investigates the asymptotic behavior of Goldbach partitions of large even integers, deriving formulas for their count and distribution, and introduces a limit theorem describing the partition size distribution as the integers grow large.

Contribution

It provides the first asymptotic formula for the number of Goldbach partitions and establishes a limit theorem for the distribution of partition sizes, using classical Tauberian methods.

Findings

Number of partitions asymptotically behaves like 2n^2 / log^2 n

Partition size distribution converges to a maximum of two independent uniform variables

Method extends to partitions into fixed numbers of prime parts

Abstract

Let $\Sigma_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We show that $\mid\Sigma_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $\Sigma_{2n}$. Let $2X_n\in (4,2n]$ be the size of this partition. We prove a limit theorem which establishes that $X_n/n$ converges weakly to the maximum of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. Our method of proof is based on a classical Tauberian theorem due to Hardy, Littlewood and Karamata. We also show that the same asymptotic approach can be applied to partitions of integers into an arbitrary and fixed number of odd prime parts