A Note on Goldbach Partitions of Large Even Integers

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Let Σ₂ₙ 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 Σ₂ₙ 2n²/²n as n→∞. We also assume that a partition is selected uniformly at random from the set Σ₂ₙ. Let 2Xₙ∈ (4,2n] be the size of this partition. We prove a limit theorem which establishes that Xₙ/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