Terence Tao described a modern proof of the prime number theorem in a lecture in UCLA, which is stated in wiki(enter link description here).

From wiki: In a lecture on prime numbers for a general audience, Fields medalist Terence Tao described one approach to proving the prime number theorem in poetic terms: listening to the "music" of the primes. We start with a "sound wave" that is "noisy" at the prime numbers and silent at other numbers; this is the von Mangoldt function. Then we analyze its notes or frequencies by subjecting it to a process akin to Fourier transform; this is the Mellin transform. Then we prove, and this is the hard part, that certain "notes" cannot occur in this music. This exclusion of certain notes leads to the statement of the prime number theorem. According to Tao, this proof yields much deeper insights into the distribution of the primes than the "elementary" proofs discussed below.

I wonder where can I find the proof? Anyone give me a reference?

  • $\begingroup$ I think this is referring to, not necessarily the paper of Riemann but roughly the same but slightly simpler using von mangoldt instead the pi functions and so on.. $\endgroup$ – sperners lemma Nov 11 '12 at 15:45
  • $\begingroup$ The only one I know is the one in Rudin's Functional Analysis, Part II. He did it as an application of distribution theory. Although he does not use Mellin transform, he uses Fourier transform and I think it is a 'modern' proof. $\endgroup$ – Hui Yu Nov 11 '12 at 15:51
  • $\begingroup$ @HuiYu. Thank you. I will read it. $\endgroup$ – hxhxhx88 Nov 12 '12 at 6:32
  • $\begingroup$ @spernerslemma. Riemann's paper provides a proof of the prime number theorem? $\endgroup$ – hxhxhx88 Nov 12 '12 at 6:33

The outline of this proof was the subject of Riemann's seminal 1859 paper, however it wasn't fully proven until Hadamard and De La Vallee Poussin in 1896. The prime number theorem is considered to be the pinnacle of 19th century number theory, and Riemmans outline motivated a lot of study in complex analysis.

One of my favorite books on analytic number theory is Montgomery and Vaughn's Multiplicative Number Theory I: Classical Theory. Try to see if you can find it in your library.

Chapter 6 deals entirely with the proof of the quantitative prime number theorem, and Chapter 5 provides some background on the Mellin transform, and Perron's formula. Of particular importance is the quantitative Perron's formula.

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  • $\begingroup$ OK. I will look at it, thanks! $\endgroup$ – hxhxhx88 Nov 14 '12 at 2:40

Personally, I prefer the proof presented here, which is a version of Newman's proof.

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    $\begingroup$ I don't think your proof is valid, look at the final statement in that proof, it doesn't make much sense. $\endgroup$ – Ethan Jan 9 '13 at 16:23

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