A Note on Goldbach Partitions of Large Even Integers

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 show that $\mid\Sigma_{2n}\mid\sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $\Sigma_{2n}$. Let $2X_n\in (4,2n]$ be the size of this partition. We prove a limit theorem which establishes that $X_n/n$ 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


Introduction and Statement of the Main Result
For a given sequence of positive integers Λ = {λ 1 , λ 2 , ...}, by a Λ-partition of the positive integer n, we mean a way of writing it as a sum of positive integers from Λ without regard to order; the summands are called parts. Let P = {p 1 , p 2 , ...} be the sequence of all odd primes arranged in increasing order. A prime partition is a Λ-partition with Λ = P. Let Q(n) be the number of prime partitions of n. Hardy and Ramanujan [6,7] were apparently the first who studied the asymptotic behavior of the number of integer (Λ = {1, 2, ...}) and prime partitions for large n. For prime partitions they proved the following asymptotic formula: log Q(n) ∼ 2π n 3 log n , n → ∞.
The study of the asymptotic behavior of Q(n) itself is quite complicated. It turns out that the corresponding asymptotic formula contains transcendental sums over the primes which can be expressed in terms of zeros of the Riemann zeta function (for more details, see e.g. [9; p. 240]). Recently Vaughan [15] proposed and studied a modification of the problem, where n is replaced by a continuous real variable. His asymptotic results avoid transcendental sums over primes.
Consider now the number Q m (n) of prime partitions of n into m parts (1 ≤ m ≤ n). The bivariate generating function of the numbers Q m (n) is of Euler's type, namely, (the proof may be found in [1; Section 2.1]). In this note we focus on the asymptotic behavior of the coefficients Q 2 (n) of x 2 and z n in the power series expansion of G(x, z) in powers of x and z. For n > 4, Q 2 (n) counts the number of ways of representing n as a sum of two odd primes. Obviously, Q 2 (n) = 0 if n is odd. In 1742 Goldbach conjectured that Q 2 (n) ≥ 1 for every even integer n ≥ 4. This problem remains still unsolved (for more details, see e.g. [8; Section 2.8 and p. 594]). Another famous conjecture related to prime partitions was stated by Hardy and Littlewood [5], who predicted the asymptotic form of Q 2 (n) for large even n. They conjectured that where C 2 is the twin prime constant In the present note we do not deal with the asymptotic equivalence (2) but consider the sum function counting all partitions of the even integers from the interval (4, 2n] into two odd prime parts. Sometimes this kind of partitions are called Goldbach partitions. Let Σ 2n denote the set of these partitions. Our main result is the following asymptotic equivalence.
Consider now a random experiment. Suppose that we select a partition uniformly at random from the set Σ 2n , i.e. we assign the probability 1/S(2n) to each Goldbach partition. We denote by P the uniform probability measure on Σ 2n . Let 2X n ∈ (4, 2n] be the number that is partitioned by this random selection. 2X n is also called the size of this partition. Using Theorem 1, we determine the limiting distribution of the random variable X n .

Remark 1.
In probabilistic terms Theorem 2 shows that the typical size of a random Goldbach partition is a fraction of 2n and X n /n converges weakly, as n → ∞, to a random variable with probability density function 2u, for 0 < u < 1, and zero elsewhere. It can be easily seen that this is the density function of the random variable U 2 , where U is a uniformly distributed random variable in the interval (0, 1).
Remark 2. One reason to study the sum function (3) is motivated by a result due to Brigham [2]. He has studied the asymptotic behavior of a similar sum function related to integer partitions weighted by the sequence of the von Mangoldt functions (the definition of a von Mangoldt function and its role in the proof of the Prime Number Theorem may be found in [8; Section 17.7]). The asymptotic behavior of a single term in Brigham's sum function was subsequently studied by Richmond [13] and Yang [16]. Their results are essentially based on Brigham's observations. Remark 3. Another interesting problem on prime partitions is related to the asymptotic behavior of the coefficients Q m (n), the number of prime partitions of n with m parts (see (1)). Haselgrove and Temperley [9; p. 240] found an asymptotic form for Q m (n), whenever m = m(n) → ∞ as n → ∞ in a proper way. In probabilistic terms their result can be stated as follows. Consider a random variable, whose probability distribution function is defined by the ratio Haselgrove and Temperley [9] showed that this random variable converges weakly to a non-degenerate random variable as n → ∞. They also determined the moment generating function of this limiting variable. The asymptotic form of the mean and the variance of probability distribution (4) were found recently by Ralaivaosaona [12]. Our paper is organized as follows. Section 2 contains some preliminaries. The proofs of Theorems 1 and 2 are given in Section 3. Our method of proof is essentially based on a classical Tauberian theorem due to Hardy, Littlewood and Karamata (see [4]).

Preliminary Results
We start with a generating function identity for the sequence {Q 2 (2k)} k>2 of the counts of Goldbach partitions.
Lemma 1 For any real variable z with | z |< 1, let Then, we have 2 Proof. Differentiating the left-hand side of (1) twice with respect to x and setting then x = 0 and m = 2, we get The last equality follows from the obvious identities Q 2 (1) = Q 2 (2) = Q 2 (4) = 0 and Q 2 (2k+1) = 0 for k = 1, 2, .... The right-hand side of (1) can be also written as exp (− p k ∈P log (1 − xz p k )). Differentiating it twice, in the same way we find that which completes the proof.
Further, we will use a Tauberian theorem by Hardy-Littlewood-Karamata whose proof may be found in [4; Chapter 7]. We use it in the form given by Odlyzko [11; Section 8.2].
Hardy-Littlewood-Karamata Theorem. (See [11; Theorem 8.7, p. 1225].) Suppose that a k ≥ 0 for all k, and that a k x k converges for 0 ≤ x < r. If there is a ρ > 0 and a function L(t) that varies slowly at infinity such that then n k=0 a k r k ∼ n r ρ L(n) Γ(ρ + 1) , n → ∞.
Remark. A function L(t) varies slowly at infinity if, for every u > 0, L(ut) ∼ L(t) as t → ∞.

Proof of the Main Result
Proof of Theorem 1. We need to show that power series (6) satisfies the conditions of Hardy-Littlewood-Karamata theorem. The next lemma establishes an asymptotic equivalence of f (z) as z → 1 − .
Lemma 2 Let f (z) be the power series defined by (5). Then, as z → 1 − , Proof. As usual, by π(y) we denote the number of primes which do not exceed the positive real number y. In (5) we set z = e −t , t > 0, and apply an argument similar to that given by Stong [14] (see also [3]). We have where I 1 (t) = t 1/2 0 π(s/t)e −s ds, I 2 (t) = ∞ t 1/2 π(s/t)e −s ds.
For I 1 (t) we use the bound π(s/t) ≤ s/t. Hence, for enough small t > 0, we obtain The estimate for I 2 (t) follows from the Prime Number Theorem with an error term given in a suitable form. So, it is known that, for y > 1, π(y) = y log y + O y log 2 y (see e.g. [10; Theorem 23, p. 65]). Furthermore, for s ≥ t 1/2 , we have log s ≥ which implies that the series k>2 Q(2k)z 2k satisfies condition (7) of Hardy-Littlewood-Karamata Tauberian theorem with r = 1, ρ = 2 and L(t) = 1 log 2 t (see also (6)). The asymptotic equivalence of Theorem 1 follows immediately from (8).
Proof of Theorem 2. Recall that 2X n ∈ (4, 2n] equals the size of a Goldbach partition that is chosen uniformly at random from the set Σ 2n of all such partitions. Since S(2n) =| Σ 2n | and, for any N ∈ Setting N ∼ un, 0 < u < 1, and applying Theorem 1 twice -to the numerator and the denominator of (14), we see that the limit of (14), as n → ∞, is u 2 . This completes the proof.