Let a > 0, b be coprime integers. Find the density of integers n for which $an + b$ is squarefree.
This question is from assignment 2 of Zeev Rudnick's Lecture notes: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html
I have read till lecture 11 but I am not sure which result I should use( relevant material can be found in Lecture 5-6) because the result over $\mathbb{Z}[x]$ are conjectures only although results over $F_{q} [t]$ are theorems.
So, can you please tell which results I should use to find density.
Let $P_a$ be the set of primes that divide $a$, and let $Q_a$ be the set of all other primes. To be square free, $an+b$ must not be divisible by $p^2$ for any prime $p$. This holds for every $p\in P_a$ by the coprimality of $a$ and $b$,, so we can focus on primes in $Q_a$. If $p \in Q_a$ then $a$ is invertible mod $p^2$, so the arithmetic progression $\{an+b : n \ge 0\}$ is uniformly distributed modulo $p^2$. Since divisibility by different primes is asymptotically independent, the limiting density of integers $n$ such that $an+b$ is square free equals $$\prod_{p \in Q_a}(1-p^{-2})= \frac{\prod_{p}(1-p^{-2})}{\prod_{p \in P_a}(1-p^{-2})}=\frac{6 }{\pi^2 \prod_{p \in P_a}(1-p^{-2})}\,$$ where the last step used the Euler product formula [1] and the value of $\zeta(2)$ [2]. See [3] for how to make the independence argument rigorous.
[1] https://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function
[2] https://en.wikipedia.org/wiki/Basel_problem
[3] https://en.wikipedia.org/wiki/Square-free_integer