Sampling Goldbach Numbers at Random

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.