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Prove the Prime Number Theorem PDF Print E-mail
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Wednesday, 17 December 2008 15:44

Let be π(x) the number of prime numbers before x (x integer).

Now, prove the following theorem.

Theorem. We have,

 π(x) ˜ x/log x

This theorem was induced by Thebychef then proved by Hadamard and De La Valle Poussin in 189. Edmund Landau and Hardy offer several elementary proof.

It is possible to prove the Theorem using Fourier Series (See Dym/Mackean - Fourier Series and Integrals, Accademic Press).

The Hint to prove this theorem is through the Riemann Zeta Function. See The Distribution of Prime Numbers by A. E. Ingham - Cambridge Tracts in Mathematics and Mathematical Physics - Cambridge University Press. 

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