Sampling Goldbach Numbers at Random

TL;DR

This paper studies the distribution of Goldbach numbers obtained by randomly partitioning even integers into two odd primes, showing their scaled mean and variance converge to specific values using generating functions and Tauberian theorems.

Contribution

It provides a new probabilistic analysis of Goldbach numbers, establishing the asymptotic mean and variance of their scaled form.

Findings

Mean of G_n/n converges to 2/3

Variance of G_n/n converges to 1/18

Method uses generating functions and Hardy-Littlewood-Karamata theorem

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 select a partition from the set $\Sigma_{2n}$ uniformly at random. Let $2G_n$ be the number partitioned by this selection. $2G_n$ is sometimes called a Goldbach number. In [6] we showed that $G_n/n$ converges weakly to the maximum $T$ of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. In this note we show that the mean and the variance of $G_n/n$ tend to the mean $\mu_T=2/3$ and variance $\sigma_T^2=1/18$ of $T$, respectively. Our method of proof is based on generating functions and on a Tauberian theorem due to Hardy-Littlewood-Karamata.