TITLE: Evaluating the sum of $kl^2$ over $p,q,k,l$ such that $pk +ql = n$
QUESTION [11 upvotes]: I have come across the following sum:
$$\sum_{\substack{p, q, k, l \in \mathbb{N} \\ k > l \\ pk + ql = n}}kl^2$$
and I am trying to simplify it, hoping to get a nice formula in terms of $n$ and some arithmetic functions of $n$.
For instance, the following sum can be evaluated as follows:
\begin{align*}
 \sum_{\substack{p, q, k, l \in \mathbb{N} \\ pk + ql = n}}kl &= \sum_{\substack{\alpha, \beta \in \mathbb{N}\\ \alpha+ \beta = n}}\sum_{k | \alpha}k \sum_{l | \beta}l \\ 
&= \sum_{\substack{\alpha, \beta \in \mathbb{N}\\ \alpha+ \beta = n}} \sigma_1(\alpha) \sigma_1(\beta)\\
&= (\sigma_1 \Delta \sigma_1)(n)
\end{align*}
where $\sigma_1(n)$ is the sum of divisors of $n$ and $\Delta$ is the discrete convolution. 
Ramanujan has a formula for the discrete convolution of $\sigma_1$ with itself given by
$$(\sigma_1 \Delta \sigma_1)(n) = \frac{5}{12}\sigma_3(n) + \frac{1}{12}\sigma_1(n) - \frac{1}{2} n \sigma_1(n)$$
where $\sigma_3(n)$ is the sum of the cubes of the divisors of $n$. 
Any thoughts on how one might proceed with the sum in the beginning of the question? The asymmetry between $k$ and $l$ is causing some problems so the same approach as for the second sum does not quite work.
I would be very much grateful for any suggestions. Thanks!

REPLY [3 votes]: Sums of this type can be calculated using standard tools from analytic number theory (Kloosterman sums + Poisson summation). This sum with additional restrictions arose in Frobenius problem. For precise statement see Lemma 8 (page 828) in the article On the Frobenius problem for three arguments (2016) by  I. Vorob'ev.
In my earlier work The solution of Arnold's problem on the weak asymptotics of Frobenius numbers with three arguments (2009) the same tools used for calculation of more simple sums
$$\sum_{\substack{p, q, k, l \in \mathbb{N} \\ pk + ql = n\\\text{+additional restrictions}}}k $$
but this technique allows to replace the summand $k$ by more or less arbitrary smooth function of $p,q,k,l.$
If you need sharper error terms then you should apply more advanced tools like cancellation of Kloosterman sums, Kuznetsov trace formula and averaging over spectrum of Laplace operator from An Additive Divisor Problem by J. M. Deshouillers and H. Iwaniec.