Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
up vote 3 down vote accepted

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

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

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

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.