Evaluate $\sum_{d\mid N}\Lambda(d)$

Question

For a positive integer $n$, define
$$\Lambda(n) = \left\{ \begin{array} {ll} \log p & \mbox{if $n = p^r$, $p$ a prime and $r \in \mathbb{N},$ }\\ 0 & \mbox{otherwise.} \end{array} \right.$$
Given a positive integer $N$, evaluate $\sum_{d\mid N}\Lambda(d)$ where the sum ranges over all divisors $d$ of $N$.

Can I get some help? I have no idea how to solve this problem

Answer 1

HINTS

1. Write $n = p_1^{e_1}p_2^{e_2} \ldots p_k^{e_k}$. What is $\Lambda (n)$?
2. What are the divisors of $n$?
3. For each divisor $d$, which have that $\Lambda(d) \neq 0$?
4. Did you know that $\log(ab) = \log a + \log b$?
5. Conclude that $\displaystyle \sum_{d \mid N} \Lambda(d) = \log N$.