On the Distribution of the Number of Goldbach Partitions of a Randomly Chosen Positive Even Integer

Let P={p1;p2;…} be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer 2k>4 is a way of writing it as a sum of two primes from P without regard to order. Let Q(2k) be the number of all Goldbach partitions of the number 2k. Assume that 2k is selected uniformly at random from the interval (4,2n], n>2, and let Yn=Q(2k) with probability 1=(n/2). We prove that the random variable Ynn/(12log⁡n)2 converges weakly, as n→∞, to a uniformly distributed random variable in the interval (0,1). The method of proof uses sizebiasing and the Laplace transform continuity theorem.