I have learnt $f(x,y)=\sum_{m,n \in \mathbb{N}}x^ny^m$ has a closed form $\frac{1}{1-x}.\frac{1}{1-y}$, but if I modified a little bit by defining a charateristic function for coprime indexes: $a(m,n)=1$ if $(m,n)=1$ and $a(m,n)=0$, otherwise, then the situation is completely different. I am completely newbie in this area and it seems there are not much research on multivariate generating functions. So my question is: Is there a closed form for the generating function as follows:
$g(x,y)=\sum_{\gcd(m,n)=1}x^ny^m$
If it exists, could you please give me the general methods to get there?
After searching for a while, I found a related open question in Does there exist a generating function for the rational numbers?
I don't believe this generating function has a closed form in any usual sense; already setting $x = y$ produces
$$f(x) = \sum_{\gcd(m, n) = 1} x^{m+n} = \sum_{n \ge 1} \varphi(n) x^n$$
(since $\gcd(m, n) = 1$ is equivalent to $\gcd(m, m+n) = 1$) and I don't believe this generating function has a closed form in any usual sense; in particular it is not rational because $\varphi(n)$ has the wrong growth rate. It's known that
$$\liminf_{n \to \infty} \frac{\varphi(n) \log \log n}{n} = e^{- \gamma}$$
but this limit is either $0$ or $\infty$ for the coefficients of a rational generating function, because the coefficients of a rational generating function necessarily grow like $Cn^r \lambda^n$ where $r$ is a non-negative integer. Actually this asymptotic behavior even implies that $f(x)$ is neither meromorphic nor algebraic.