In this answer it uses $\lvert\cdot\rvert$ but I can't deduce its meaning from the context.
I'm referring to the expression:
$$|f^N(n)|\le |F^N(n)| \le \frac{|n|+N}{N}$$
We have non-negative integers $n$ so $\lvert n\rvert=-n$ if $n$ is negative doesn't entirely jump out.
A metric of sorts $\lvert n\rvert=N^{-\nu_N(x)}$ would make sense but then there are two valuations $\nu_N$ and $\prod_{j\le J} p_j^{-v_{p_j}(n)}$ to choose from and I can't tell from the context which of these three is intended.