Question about the prime number theorem

In summary, the prime number theorem is a mathematical theorem that states the asymptotic distribution of prime numbers. It was first discovered by mathematicians Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896. It has numerous applications in mathematics, including in number theory, algebra, and analysis, as well as in cryptography and coding theory. The theorem has been proven using complex analysis and the theory of functions of a complex variable. There are no exceptions to the theorem, but it is an asymptotic result that only holds true for large values of n.
  • #1
AxiomOfChoice
533
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!
 
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  • #2
You have that p_n = (1 + o(1))(n log n). Take the log of both sides and rewrite as a limit.
 

1. What is the prime number theorem?

The prime number theorem is a mathematical theorem that states the asymptotic distribution of prime numbers. It shows that the number of primes less than or equal to a given number is approximately equal to the ratio of that number to its natural logarithm.

2. Who discovered the prime number theorem?

The prime number theorem was first discovered by mathematicians Jacques Hadamard and Charles Jean de la Vallée Poussin independently in 1896.

3. How is the prime number theorem used in mathematics?

The prime number theorem has numerous applications in mathematics, including in the study of number theory, algebra, and analysis. It is also used in cryptography and coding theory.

4. Is the prime number theorem proven?

Yes, the prime number theorem has been proven using complex analysis and the theory of functions of a complex variable.

5. Are there any exceptions to the prime number theorem?

No, the prime number theorem holds true for all positive integers. However, it is an asymptotic result, meaning that it only holds true for large values of n.

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