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Although i am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was G.H.Hardy so much keen to have an elementary proof of the Prime Number Theorem. He also stated that producing such a proof, will change the complexion of Mathematics, but nothing like that has happened. What was on Hardy's mind?

The elementary proof although has some intricate tricks involved, i am curious to know whether the methodology can be applied for attacking more complex problems. I have seen that the analytic proof has a continuation and is not over and discuss some more interesting properties regarding the $\zeta$ function.

I also saw this thread. Are there any such theorem's which piques peoples curiosity in getting an elementary proof.(While writing this FLT comes to my mind!!).

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This should be community wiki. –  Tom Stephens Aug 15 '10 at 19:17
    
What do you mean by "posing a serious threat to Mathematical community"? (Did you mean something like "posing a serious challenge", perhaps?) –  ShreevatsaR Aug 15 '10 at 19:19
    
@Tom Stephens: Whats community wiki :x) –  anonymous Aug 15 '10 at 19:26
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I've always thought number theorist's notions of what is "elementary" is not were strange. Here's a related discussion on Math Overflow. mathoverflow.net/questions/4463/… –  John D. Cook Aug 15 '10 at 19:26
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Chandru: Hardy thought a proof avoiding complex analysis would be impossible, based on how he understood the zeta-function, so that if such a proof were found it would mean people didn't understand the subject correctly. This was said before such a proof was found. Once such a proof was found, well, frankly there were no fundamental new consequences. Hardy was just wrong. The end. –  KCd Aug 16 '10 at 2:07

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up vote 19 down vote accepted

As KCd explains in a comment, the proof of the PNT in Hardy's time seemed to be intimately connected to the complex analytic theory of the $\zeta$-function. In fact, it was known to be equivalent to the statement that $\zeta(s)$ has no zeroes on the half-plane $\Re s \geq 1$. Although this equivalence may seem strange to someone unfamiliar with the subject, it is in fact more or less straightforward.

In other words, the equivalence of PNT and the zero-freeness of $\zeta(s)$ in the region $\Re s \geq 1$ does not lie as deep as the fact that these results are actually true.

The possibility that one could then prove PNT in a direct way, avoiding complex analysis, seemed unlikely to Hardy, since then one would also be giving a proof, avoiding complex analysis, of the fact that the complex analytic function $\zeta(s)$ has no zero in the region $\Re s \geq 1$, which would be a peculiar state of affairs.

What added to the air of mystery surrounding the idea of an elementary proof was the possibility of accessing the Riemann hypothesis this way. After all, if one could prove in an elementary way that $\zeta(s)$ was zero free when $\Re s \geq 1$, perhaps the insights gained that way might lead to a proof that $\zeta(s)$ is zero free in the region $\Re s > 1/2$ (i.e. the Riemann hypothesis), a statement which had resisted (and continues to resist) attack from the complex analytic perspective.

In fact, when the elementary proof of PNT was finally found, it didn't have the ramifications that Hardy anticipated (as KCd pointed out in his comment).

For a modern perspective on the elementary proof, and a comparison with the complex analytic proof, I strongly recommend Terry Tao's exposition of the prime number theorem. In this exposition, Tao is not really concerned with elementary vs. complex analytic techniques as such, but rather with explaining what the mathematical content of the two arguments is, in a way which makes it easy to compare them. Studying Tao's article should help you develop a deeper sense of the mathematical content of the two arguments, and their similarities and differences.

As Tao's note explains, both arguments involve a kind of "harmonic analysis" of the primes, but the elementary proof works primarily in "physical space" (i.e. one works directly with the prime counting function and its relatives), while the complex analytic proof works much more in "Fourier space" (i.e. one works much more with the Fourier transforms of the prime counting function and its relatives).

My understanding (derived in part from Tao's note) is that Bombieri's sieve is (at least in part) an outgrowth of the elementary proof, and it is in sieve methods that one can look to find modern versions of the type of arguments that appear in the elementary proof. (As one example, see Kevin Ford's paper On Bombieri's asymptotic sieve, which in its first two pages includes a discussion of the relationship between certain sieving problems and the elementary proof.) But I should note that modern analytic number theorists don't pursue sieve methods out of some desire for having "elementary" proofs. Rather, some results can be proved by $\zeta$- or $L$-function methods, and others by sieving methods; each has their strengths and weaknesses. They can be combined, or played off, one against the other. (The Bombieri--Vinogradov theorem is an example of a result proved by sieve methods which, as far as I understand, is stronger than what could be proved by current $L$-function methods; indeed, it an averaged form of the Generalized Riemann Hypothesis.)

To see how this mixing of methods is possible, I again recommend Tao's note. Looking at it should give you a sense of how, in modern number theory, the methods of the two proofs of PNT (elementary and complex analytic) are not living in different, unrelated worlds, but are just two different, but related, methods for approaching the "harmonic analysis of the primes".

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Nice answer! But you didn't answer the last part of my question: Are there any such theorem's which piques peoples curiosity in getting an elementary proof.(While writing this FLT comes to my mind!!).? –  anonymous Aug 16 '10 at 4:36
    
Matt, the link to Terry's page has the ~ missing, so it doesn't work. –  KCd Aug 16 '10 at 5:50
    
Dear Keith, Thanks for spotting this; it is now fixed. –  Matt E Aug 16 '10 at 6:36

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