I am counting the number of unique polynomial candidates for a fixed $r$ where $1 \le r \le N$ with $|s|, |t| \le N$ for naive height $N \ge r$. This sum is $${T}_{2} \left({r, N}\right) = \sum_{\substack{- N \le s, t \le N \\ \left({r, s, t}\right) = 1}} 1$$ From Randell Heyman and Igor E. Shparlinski "On the Number of Eisenstein Polynomials of Bounded height" defines a generalization of the Euler function $$\varphi \left({r, N}\right) = \sum_{\substack{- N \le s \le N \\ \left({r, s}\right) = 1}} 1$$
We can further generalize this with two summation parameters as $${\varphi}_{2} \left({r, N}\right) = \sum_{\substack{- N \le s, t \le N \\ \left({r, s, t}\right) = 1}} 1$$ Following R Heyman's paper and applying the Principle of Inclusion-Exclusion I can write $${T}_{2} \left({r, N}\right) = {\varphi}_{2} \left({r, N}\right) = \sum_{d \mid r} \mu \left({d}\right) \sum_{\substack{- N \le s \le N \\ d \mid s}} 1 \sum_{\substack{- N \le t \le N \\ d \mid t}} 1 = \sum_{d \mid r} \mu \left({d}\right) \left({2\, \lfloor{\frac{N}{d}}\rfloor + 1}\right)^{2}.$$ where $\mu \left({d}\right)$ is Mobius function.
I wish to take the limit as $N \rightarrow \infty$ resulting in $$\lim_{N \rightarrow \infty} {T}_{2} \left({r, N}\right) = \sum_{d \mid r} \mu \left({d}\right) \left({2\, \frac{N}{d} + \mathcal{O} \left({1}\right)}\right)^{2} = 4\, {N}^{2} \sum_{d \mid r} \frac{\mu \left({d}\right)}{{d}^{2}} + 4\, N \sum_{d \mid r} \frac{\mu \left({d}\right)}{d} + \mathcal{O} \left({\sum_{d \mid r} \mu \left({d}\right)}\right).$$ I can account for all the sums except $$\sum_{d \mid r} \frac{\mu \left({d}\right)}{{d}^{2}} = ???$$.
So is this function ${\varphi}_{2} \left({r, N}\right)$ a known function, is this generalization correct, and if so I am looking for a reference or so and also I need to find the above sum.
Obviously ${T}_{2} \left({r, N}\right)$ is the generalization of ${\varphi} \left({r, N}\right)$ so no need to give it a difference name here. Now note that from the Mobius inversion we have $$\sum_{d \mid r} \frac{\mu \left({d}\right)}{{d}^{2}} = \prod_{p \mid r} \left({1 - \frac{1}{{p}^{2}}}\right) = \frac{{J}_{2}\left({r}\right)}{{r}^{2}}$$ where ${J}_{k} \left({n}\right)$ is Jordan Totient Function. Using other well known identities, I can write $$\lim_{N \rightarrow \infty} {T}_{2} \left({r, N}\right) \sim 4\, {N}^{2}\, \frac{{J}_{2}\left({r}\right)}{{r}^{2}} + 4\, N\, \frac{{\varphi} \left({r}\right)}{r} + \mathcal{O} \left({{2}^{\omega \left({r}\right)}}\right) \sim 4\, {N}^{2}\, \frac{{J}_{2}\left({r}\right)}{{r}^{2}} + \mathcal{O} \left(N{}\right)$$ with $$\sum_{d \mid r} \mu \left({d}\right) = {2}^{\omega \left({r}\right)}$$ where $\omega \left({r}\right)$ is the number of distinct prime factors of $r$.