Can someone find a function f(n) satisfying these bounds? Can you also prove that it does?
$$ \sum\limits_{k=1}^n \Lambda(k) [1-\text{Frac}(\frac{n}{k})][1-\frac{k}{n}\text{Frac}(\frac{n}{k})]=\frac{1}{2}\sum\limits_{k=1}^n \Lambda(k){}\text{}+O(f(n))$$
and
$$ f(n) = o(n)$$,
where $\displaystyle \text{Frac}(\frac{n}{k})$ is the fractional part of $\displaystyle \frac{n}{k}$ and $\Lambda(k)$ is the von Mangoldt function. I would greatly appreciate any help, if someone could even give me an elementary proof I would be willing to do something for them in return.
