# Pi[x] >=loglogx

1. ### AlbertEinstein

113
I am going through Hardy's book on number theory.The following theorem I do not understand.

theorem 10: pi[x] >= loglog x
where pi[x] is the prime counting function
and >= stands for greater than or equal to

The arguments written in the book are very compact.please help .

2. ### shmoe

1,994
Do you follow any of it?

Do you understand how they derived $$p_{n}<2^{2^n}$$ ?

this is an important step. The rest just follows from pi(x) being increasing, and also $$\pi(p_n)=n$$ which they use but don't explicitly mention.

3. ### CRGreathouse

3,497
Bertrand's postulate?

4. ### shmoe

1,994
Nope! the bound of p_n above is much weaker than Bertrand's will give you. It's correspondingly simpler to prove though, it follows from a slight adaptation of Euclid's proof there are infinitely many primes (in case anyone who hasn't seen it wants to give it a stab)

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