Question about the prime number theorem

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SUMMARY

The discussion focuses on proving the limit statement regarding the prime number theorem: \(\lim_{n\to \infty} \frac{\log (\log p_n)}{\log n} = 0\). Participants are allowed to assume the established result \(\lim_{n\to \infty} \frac{p_n}{n \log p_n} = 1\). The conversation emphasizes the relationship between prime numbers and logarithmic functions, specifically utilizing the asymptotic expression \(p_n = (1 + o(1))(n \log n)\) to derive the limit through logarithmic manipulation.

PREREQUISITES
  • Understanding of prime number notation and properties, specifically \(p_n\)
  • Familiarity with limits and asymptotic notation, particularly \(o(1)\)
  • Knowledge of logarithmic functions and their properties
  • Basic concepts of calculus, particularly limits and continuity
NEXT STEPS
  • Study the derivation of the prime number theorem and its implications
  • Explore the properties of logarithmic functions in asymptotic analysis
  • Learn about the implications of the result \(\lim_{n\to \infty} \frac{p_n}{n \log p_n} = 1\)
  • Investigate advanced topics in analytic number theory related to prime distributions
USEFUL FOR

Mathematicians, students of number theory, and anyone interested in the analytical properties of prime numbers and their distributions.

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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!
 
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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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