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Question about the prime number theorem

  1. Apr 27, 2010 #1
    Let [itex]p_n[/itex] be the nth prime number. Can someone help me figure out how to show that

    [tex]
    \lim_{n\to \infty} \frac{\log (\log p_n)}{\log n} = 0.
    [/tex]

    You're allowed to assume that

    [tex]
    \lim_{n\to \infty} \frac{p_n}{n \log p_n} = 1.
    [/tex]

    I'm quite confident what I want to show is true, but it's hard to figure out how to do it because [itex]p_n > n[/itex] for every n. Thanks!
     
  2. jcsd
  3. Apr 27, 2010 #2

    CRGreathouse

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    Science Advisor
    Homework Helper

    You have that p_n = (1 + o(1))(n log n). Take the log of both sides and rewrite as a limit.
     
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