Alternative way to compute some sums from number theory.

Question

Let $*$ stands for Dirichlet's convolution operator

I know that sum of the form

1. $$\sum_{1\le n\le x}(f*g)(n)$$

is somehow easy to compute/estimate using methods such:

• changing order of summation

• using Dirichlet's hyperbola method.

But what if we consider sum of the form

2. $$F(N):=\sum_{1\le n \le N}(f*g)(n)\cdot(f*g)(N-n)$$

It is getting harder to interpret. I see that this is Cauchy convolution of the same function which is Dirichlet convolution.

In first case we were able to draw an area bounded by hyperbola and then count values on points in it using different way.

But in second case we should consider all the values from the set $$A:=\{x,y,z,t\in\{1,...,N\}:x\cdot y+z\cdot t \le N\}$$ and change order of taking their values. But it is hard (for me) to imagine how this set looks like.

I have a question:

How to rewrite $F(N)$ so that we could tell about it something new? Is there an extended versions of hyperbola method?

Also i have an idea to associate $F(N)$ some generating functions as it is a coefficient of products of other generating functions. But the problem is to find a common language of Dirichlet series and power series.

Is this true that set A has three dimensions? If so, then how it looks like?

Regards.