If $n$ is a positive integer, let $r(n)$ denote the number of representations of $n$ as a sum of products of pairs of positive integers. (Here, the order of the terms in the sum does not matter, but products with the same answer are regarded as different, even if they contain the same two numbers in a different order.) For example,

r(1) = 1;

r(2) = 3:

2 = 2*1 = 1*2 = 1*1 + 1*1;

r(3) = 5:

3 = 3*1 = 1*3 = 2*1 + 1*1 = 1*2 + 1*1 = 1*1 + 1*1 + 1*1.

I would very much like to know whether the asymptotic of $r(n)$ is known, or whether it can be derived easily from known results/methods.

(Of course, the problem can be rephrased in terms of $\Lambda$-partitions. Recall that if $\Lambda$ is a non-decreasing sequence of positive integers, $$\Lambda_1 \leq \Lambda_2 \leq \ldots,$$ with $\Lambda_k \to \infty$ as $k \to \infty$, a $\Lambda$-partition of $n$ is a representation of $n$ as a sum $$n = \sum_{k=1}^{\infty} m_k \Lambda_k,$$ with $m_k \in \mathbb{N} \cup \{0\}$ for all $k$. If $\Lambda$ is the sequence with $d(s)$ copies of $s$ for every positive integer $s$, where $d(s)$ denotes the number of divisors of $s$, then $r(n)$ is the number of $\Lambda$-partitions of $n$.)

The generating function of $r(n)$ is given by

$$\sum_{n=0}^{\infty} r(n) z^n = \prod_{(k,l) \in \mathbb{N}^2} (1-z^{kl})^{-1},$$

and so the theorem of N. A. Brigham in [A General Asymptotic Formula for Partition Functions, Proc. Amer. Math. Soc., 1950] gives the asymptotic of $\log r(n)$:

$$\log r(n) \sim \sqrt{2\zeta(2) n \log n} = \pi \sqrt{\tfrac{1}{3} n \log n}.$$

Unfortunately, the more precise Tauberian theorem of Ingham [A Tauberian Theorem for Partitions, Ann. Math, 1941], which gives the asymptotic of $a(n)$ for a wide class of partition functions $a(n)$, does not seem to be applicable, since his function $R(u)$ (in his Theorem 2) grows like $u \log u$ in the above case, rather than like $u^{\beta}$+(small error) for some fixed $\beta >0$. The associated Dirichlet series

$$D(s) = \sum_{(i,j)\in \mathbb{N}^2} s^{-ij} = \zeta(s)^2$$

is the square of the Riemann zeta function. This has a pole of order 2 at $s=1$, so the theorem of Meinardus on $\Lambda$-partitions is not applicable, either.

I am not familiar enough with the area to know whether the Hardy-Littlewood circle method, or some other method, can be adapted to work in the case of $r(n)$. Any help would be greatly appreciated!

b for a and b some numbers. If ab*c is allowed, then I don't know how to approach r(n) at all. $\endgroup$ – The Masked Avenger Mar 10 '14 at 17:07