I came across the following definitons:
Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension $\mathbb{Q}_p(\zeta_n)/\mathbb{Q}_p$ be $d_p$. Define $D$ to be the set of positive integers $m$ such that $d_p\mid v_p(m)$ for all prime $p\nmid n$ and $v_p(m)=0$ for all primes $p\mid n$, where $v_p(m)$ is the valuation function. For $m\in D$, write $ m=\prod_{i-1}^kp_i^{t_i}$ and define $\Psi$ as $$\Psi(m)= \prod_{i-1}^kp_i^{(d_{p_i}-1)t_i/d_{p_i}}$$ we let it equal zero if $m\notin D$.
Here's my question:
What is the intuition behind the definition of $D$ and $\Psi$? For example I mean if $n=2$ I can see that only the odd integers are in $D$ and $\Psi(x)=1$ since $d_p=1$ for all $p$. But what about in general? What is in the set $D$ and what is the map $\Psi$?