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Any hints how to prove for $n \in \mathbb N$ $$ \frac{1}{n} \sum_{p \in \mathbb P} \left\lfloor \frac{n}{p} \right\rfloor \log p = \log n + O(1) $$ where $\mathbb P$ denotes the set of all primes? As for all primes $p > n$ we have $\lfloor n/p \rfloor = 0$, the sum is finite, i.e. if $p_1, \ldots, p_k$ are all primes $\le n$ we have $$ \frac{1}{n} \sum_{p \in \mathbb P} \left\lfloor \frac{n}{p} \right\rfloor \log p = \frac{1}{n} \sum_{i=1}^k \left\lfloor \frac{n}{p_i} \right\rfloor \log p_i = \frac{1}{n} \sum_{i=1}^k \log\left( p_i^{\lfloor n/p_i \rfloor}\right) = \frac{1}{n} \log\left( \prod_{i=1}^k p_i^{\lfloor n/p_i\rfloor} \right) $$ so I guess we should somehow show that $\log n^n \approx \log\left( \prod_{i=1}^k p_i^{\lfloor n/p_i\rfloor} \right)$, but I have no idea how to proceed, so any hints?

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2 Answers 2

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Consider that:

$$ n! = \prod_{p\leq n} p^{\left\lfloor\frac{n}{p}\right\rfloor+\left\lfloor\frac{n}{p^2}\right\rfloor+\left\lfloor\frac{n}{p^3}\right\rfloor+\ldots}$$ hence: $$ \log(n!) = \sum_{p\leq n}\left\lfloor\frac{n}{p}\right\rfloor \log p + \sum_{k\geq 2}\sum_{p\leq n}\left\lfloor\frac{n}{p^k}\right\rfloor \log p = \sum_{p\leq n}\left\lfloor\frac{n}{p}\right\rfloor \log p + O(n)$$ and the LHS can be estimated through Stirling's approximation.

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  • $\begingroup$ The assertion that the second part is $O(n)$ requires an argument, but yes. :) $\endgroup$ Jun 19, 2015 at 16:33
  • $\begingroup$ I found that for $k \ge 2$ we have $\sum_{p \le n} \lfloor n / p^k \rfloor \log p \in O(n)$, but that the infinite sum over all $k \ge 2$ is in $O(n)$ I did not see, would you mind giving another hint? $\endgroup$
    – StefanH
    Jun 19, 2015 at 16:41
  • $\begingroup$ Okay, I see it, because the sum is finite. $\endgroup$
    – StefanH
    Jun 19, 2015 at 16:55
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We have $$\frac{1}{n}\sum_{p\leq n}\left\lfloor \frac{n}{p}\right\rfloor \log\left(p\right)=\sum_{p\leq n}\frac{\log\left(p\right)}{p}+O\left(\frac{1}{n}\sum_{p\leq n}\log\left(p\right)\right) $$ because $\left\lfloor \frac{n}{p}\right\rfloor =\frac{n}{p}+O\left(1\right) $ and by Mertens first theorem we have that $$\sum_{p\leq n}\frac{\log\left(p\right)}{p}=\log\left(n\right)+O\left(1\right) $$ and by PNT $$\sum_{p\leq n}\log\left(p\right)=O\left(n\right) $$ hence $$\frac{1}{n}\sum_{p\leq n}\left\lfloor \frac{n}{p}\right\rfloor \log\left(p\right)=\log\left(n\right)+O\left(1\right). $$

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    $\begingroup$ Thanks for your answer too! $\endgroup$
    – StefanH
    Jun 19, 2015 at 17:05

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