Can someone find a function f(n) satisfieing 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)),\text{ Such that:}\lim_{n\to\infty}f(n)/n=0$
Where $\displaystyle \text{Frac}(\frac{n}{k})$ is the fractional part of $\displaystyle \frac{n}{k}$, and where $\Lambda(k)$ is the Von-Mangoldt function. I know that a function does exist, I just cant prove that it does. I would Greatly appreiciate any help though, and if someone could even give me an elementary proof I would be willing to do somthing for them in return.