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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.

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I doubt there is a reasonable elementary proof. Note that the left-hand sum equals $$ \sum_{F=1}^n \sum_{n/(F+1)

However, I don't know how to proceed further without plugging in the Prime Number Theorem, say in the form $|E(x)| \le C(A) x/(\log x)^A$ for any $A>0$ and some constant $C(A)$ depending on $A$.

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    If I could reduce the prime number theorem to this problem, do you think it would be worth pursuing? Also are Riemann-Stieltjes integrals considered elementary?2012-10-26
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    Personally, I think this problem is slightly harder than the prime number theorem (where the summand is simply $\Lambda(k)$). So reducing PNT to this problem isn't likely to be fruitful in my opinion. Riemann-Stieltjes integrals are more or less as elementary as regular Riemann integrals - they're just less common. The good news is, the formula I derived using R-S integrals can be verified using nothing more than first-year calculus. (Also, "elementary" isn't a firm label, so deciding whether a method is or isn't elementary isn't that mathematically significant.)2012-10-26
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    What is meant by the letter F? does that denote the fractional part of n/k? or what?2012-10-31
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    $F$ is simply an index of summation (see the $\sum_{F=1}^n$?). In the first step, it equals the integer part of $n/k$; thereafter, it's simply an integer parameter.2012-11-03