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

2016/02/03 by Mutafchiev, Ljuben
#11P32 #60C05 #60F05 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1602.01232

Abstract

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 Y_n=Q(2k) with probability 1/(n-2). We prove that the random variable \fracY_nn/((1)/(2)logn)² converges weakly, as n→∞, to a uniformly distributed random variable in the interval (0,1). The method of proof uses size-biasing and the Laplace transform continuity theorem.

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