My lecturer went through the method but I would like it to be cleared up,
$\sigma(n)= \sum_{d|n}d,$ find the generating function. he defined $f(n)=n$ and $g(n)=1$.
Then,$$\sigma(n)= \sum_{d|n} f(d)g(\frac{n}{d}),$$ where $$f(n)= \sum_{n=1}^{\infty}\frac{n}{n^s}= \sum_{n=1}^{\infty}\frac{1}{n^{s-1}}=\xi(s-1).$$ Next, $$\sum_{n=1}^{\infty}\frac{g(n)}{n^s}=\sum_{n=1}^{\infty}\frac{1}{n^s}=\xi(s),$$ thus $$\sum_{n=1}^{\infty}\frac{\sigma (n)}{n^s}=\xi(s)\xi(s-1).$$
I do not know how he chose the two functions at the beginning nor the steps he took to deduce the method please may someone clear this up.