Sum involving integer part and cosine function

Question

How to find the close form of sum and eliminate $k$?

$$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$

Answer 1

In the following, let $\lfloor x\rfloor$ be the largest integer not greater than $x$.

Since $0\le x-\lfloor x\rfloor\lt 1$ holds for any $x\in\mathbb R$, we have $$0\le \frac nk-\left\lfloor\frac nk\right\rfloor\lt 1\Rightarrow (0\lt) \cos 1\lt \cos\left(\frac nk-\left\lfloor\frac nk\right\rfloor\right)\le \cos 0=1.$$

This leads that $$\left\lfloor\cos\left(\frac nk-\left\lfloor\frac nk\right\rfloor\right)\right\rfloor=\begin{cases}1&\text{if $k$ is a positive divisor of $n$}\\0&\text{otherwise}\end{cases}$$

Hence, the sum will be $$\sum_{\text{$k$ is a positive divisor of $n$}}\frac{n}{k}=\text{the sum of the positive divisors of $n$}.$$