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Can someone help me prove the inequality, $$ \frac{\sum\limits_{k=1}^n \Lambda(k) \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n \Lambda(k)}<\ \frac{\sum\limits_{k=1}^n 1* \frac{k}{n}\lceil\frac{n}{k}\rceil\{ \frac{n}{k} \}}{\sum\limits_{k=1}^n 1}$$

and integrate the following function from x=0 to x=1 $$ x\lceil\frac{1}{x}\rceil\{ \frac{1}{x} \}$$

Where { n/k } is the fractional part of n/k , $ \lceil{n}/{k}\rceil{} $ is the ceiling function applied to n/k, and where $\Lambda(k)$ is the Von-Mangoldt function.

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