For any positive $x$, we have
$$ \sum_{n \leq x} \sum_{d | n} \varLambda(d) = \sum_{n \leq x} \varLambda(n)\lfloor\ x/n\rfloor, $$ where $\varLambda$ is the von-Mangoldt function.
Now, let $K$ denote a number field of degree $n$ and $\mathcal{O}_K$ be the ring of integers of $K$. It is a fact that every non-zero proper ideal $\mathfrak{a} \subset \mathcal{O}_K$ can be written as a product of prime ideals in a unique way. The norm of an ideal $\mathfrak{a} \subset \mathcal{O}_K$, denoted by $N(\mathfrak{a})$, is defined as the cardinality of the quotient $\mathcal{O}_K / \mathfrak{a}$.
Can we see the following equality which is an analogue of the above? $$ \sum_{N(\mathfrak{a}) \leq x} \sum_{\mathfrak{b} | \mathfrak{a}} \varLambda(\mathfrak{b}) = \sum_{N(\mathfrak{a}) \leq x} \varLambda(\mathfrak{a})\lfloor\ x/N(\mathfrak{a})\rfloor $$