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So this question has been asked before, see here, but instead of how to go from part 4 to part 5, I am having a difficult time proving part 4:

For each $\alpha > 0$ there exists a sequence of integers $\{n_1, n_2, \dots \}$, increasing in the weak sense, such that $p_{n_j} \sim \alpha j \qquad (j \to \infty)$.

How does it follow from the previous part?

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A first step should be to figure out what the $n_j$ should look like. Because $p_n\sim n\log n$, we need that the sequence satisfies $\alpha j \sim n_j \log n_j$. Note that if $n_j \sim \alpha j/\log j$, then $\log n_j \sim \log \alpha +\log j -\log \log j \sim \log j$. Is this precise enough? Why or why not? –  Aaron Sep 25 '11 at 5:04
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up vote 3 down vote accepted

Consider $$n_j=\left[\frac{\alpha j}{\log j}\right].$$

Then $$p_{n_j}\sim n_j \log(n_j)\sim \frac{\alpha j}{\log j}\log\left(\frac{\alpha j}{\log j}\right)=\alpha j+\frac{\alpha j}{\log j}\log\left(\frac{\alpha}{\log j}\right)$$

$$=\alpha j+O\left(\frac{\alpha j\log \log j}{\log j}\right)\sim \alpha j.$$

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+1 for the correct use of several signs $\sim$ and $=$ in the same sentence. –  Did Sep 25 '11 at 17:06
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