Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}. 
$$
I assume there must be something known about this quantity... If someone could provide me a reference it would be appreciated. Thank you


ps I assume $\deg f > 0$.



Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
The assumption that $f(X)$ and $g(X)$ are relatively prime means that there is
a positive integer $R\_{f,g}=\operatorname{Resultant}(f,g)$ and polynomials $a(X)$ and $b(X)$ in $\mathbb Z[X]$ so that
$$ a(X)f(X) + b(X)g(X) = R\_{f,g}. $$
Hence if $x\in\mathbb Z$ satisfies $f(x)\mid g(x)$, then $f(x)\mid R$. Since $\bigl|f(x)\bigr|\to\infty$ as $|x|\to\infty$, this shows that your quantity $N(B)$ is bounded as $B\to\infty$. A relatively trivial bound is simply
$$ N(B) \le \#\Bigl\{x\in\mathbb Z : \bigl| f(x) \bigr| \le R\_{f,g} \Bigr\}. $$
One can bound $R\_{f,g}$ in terms of the coefficients of $f$ and $g$ and their degrees, if you want a more explicit bound.


