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The Prime Number Theorem was originally proved using methods in complex analysis. Erdos and Selberg gave an elementary proof of the Prime Number Theorem. Here, "elementary" means no use of complex function theory.

Is it possible that any theorem in number theory can be proved without use of the complex numbers?

On the one hand, it seems a lot of the theorems using in analytic number theory are about the distributions of primes. Since the Prime Number Theorem has an elementary proof, this might suggest that elementary proofs exist in other cases.

On the other hand, the distribution of primes is intimately related to the zeros of the Riemann Zeta function. Perhaps the proofs of other statements in analytic number theory require more direct references to the Riemann Zeta function.

This topic is more of a fascination for me, as I am not a number theorist. I would be interested if there are other examples of elementary proofs of theorems originally proved with complex analytic methods.

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Although you're not a number theorist, can you tell us what you are? This may be useful in giving answers that are meaningful to you. –  KConrad Aug 22 '10 at 22:26
    
Grad student in algebraic topology. Have fond memories of my undergraduate number theory course; maybe I was feeling nostalgic today. –  Micah Miller Aug 23 '10 at 0:05
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The following answer to a related question on the elementary proof of the PNT may be of interest: math.stackexchange.com/questions/2530/… –  Emerton Aug 23 '10 at 0:12

7 Answers 7

up vote 14 down vote accepted

Yes, there is a theorem to this effect by Takeuti given in his book "Two applications of logic to mathematics". He shows roughly that complex analysis can be developed in a conservative extension of Peano arithmetic.

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Yes - but I don't think that's the kind of thing people have in mind when they talk of replacing complex analysis with elementary arguments. I'm pretty sure that what Erdos and Selberg did had nothing to do with developing Hadamard's proof in a conservative extension of Peano arithmetic. –  Gerry Myerson Aug 22 '10 at 23:21
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Right, but it's going to be very hard to make mathematically precise the notion of giving an elementary proof which doesn't include the proofs you can construct via Takeuti's results. –  Noah Snyder Aug 23 '10 at 4:23
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@Noah, of course, you are correct, but in an abstractly similar situation US Supreme Court Justice Potter Stewart said "I know it when I see it." –  Gerry Myerson Aug 23 '10 at 5:37
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My understanding is that given Hadamard's proof (which uses complex analysis) and Takeuti's result you could posit the existence of, and even construct, an elementary proof - but said elementary proof would not look anything like the Erdos-Selberg proof. Takeuti tells you how to turn an elevator into a staircase, whereas Erdos-Selberg build a staircase from scratch. (Anyone have a better metaphor?) –  Gerry Myerson Aug 24 '10 at 0:43
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@Gerry: It is my impression that the Erdos-Selberg proof doesn't build the staircase from scratch but rather uses motivations from complex analysis in finding certain formulas expressible in an elementary way. It isn't a mechanical process from the complex to the real, but there is analogy involved. –  Pace Nielsen Aug 24 '10 at 18:01

I just heard about a very interesting sounding new approach to analytic number theory by A. Granville and K. Soundararajan, which seems to fit to your question: "Since 1859 the only coherent approach to these problems has been based on Riemann's idea connecting the distribution of prime numbers to the zeros of the Riemann zeta function -- which are the zeros of an analytic continuation. Some might argue that this is "unnatural" and ask for an approach that is less far removed from the original problems. Recently Soundararajan and I have proposed a different approach to the whole subject of analytic number theory, based on our concept of pretentiousness -- recently we have realized our dream of being able to develop the whole subject in a coherent way, without using the zeros of the Riemann zeta function."(link to course website, course notes) Perhaps someone tells us more about it?

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Here is a recent paper by Dimitris Koukoulopoulos, which obtains the strongest known form of the Prime Number Theorem without heavy use of complex analysis. One can find the symbol $i$ in the paper, but it does not rely on the analytic continuation of the zeta function.

This is a direct extension of the "pretentious" methods of Granville and Soundararajan, mentioned in an answer by Thomas Riepe.

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It depends on what one would count as "number theory". Even if one´s interest ultimately lies "only" in algebraic number subfields of the reals, one could argue that there are objects attached to these structures (e.g. L- and Zeta-functions) that are number-theoretical and are of interest for themselves, that is, they are more than just a way prove other things. I think a nice example here is Artin´s conjecture on the holomorphy of the Artin´s L-function -> http://en.wikipedia.org/wiki/Artin_L-function. To elaborate a little bit on this, there is absolutely no reason why an arbitrary ordinary Dirichlet series should admit meromorphic continuation, let alone being entire.

Assuming that the purely algebraic and real-analytic approach turns out to suffice for almost all number-theorerical purposes, objects like the complex-valued L-functions may indeed not contribute so much more to our mathematical understanding, but they certainly do so to our "philosphical" (or if you want "meta-mathematical") understanding.

Moreover, in the end one of the things that matter are the interconnections between various fields of mathematics. In that case one a priori has to deal with structures even "higher" than the complex numbers, e.g. automorphic forms -> http://en.wikipedia.org/wiki/Langlands_program.

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Although stuffs like the prime number theorem or Dirichlet's theorem on primes in arithmetic progression are believed to form the integral parts of analytic number theory, I have something to say, that might interest you. There is a very intuitive way of guessing the asymptotic expression x/logx for pie(x). I was going through the book "What is Mathematics" by Courant, Robbins and Stewart a few days back, from where I got this idea. It involves expressing log n!in two ways, one by its asymptotic expression nlogn and the other involves De-Polygnac's theorem on the greatest power of a prime that divides n!.Then the two expressions are compared. The next step involves dividing the interval [2,x] into a large number of sub-intervals where #primes is approximated by P(x)dx. This P(x) is actually assumed to be a smooth prime density function whose definite integral gives the value of pie(x). The comparison yields an asymptotic expression for P(x), namely
(x-1)/(xlogx) whose integral is approximately x/logx for large x. However, as mentioned there, the primary difficulty behind the proof is really the existence of such a smooth density function P(x). This argument may seem to be a more statistical in nature! You can refer to Hardy's work on the zeros of the Riemann Zeta function. Another fact that might interest you as well, a theorem that states that for any k, there are k collinear points on the prime number graph. You can consult Carl Pomerance's paper for a proof, which uses the concept of convex hull.

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There are a lot of interesting facts in mathematics, but we're looking for answers to a specific question here, and I don't see how Pomerance's paper qualifies. –  Gerry Myerson Sep 2 '12 at 5:32
    
accidentally flagged Gerry's comment while intending to +1 it :-( –  Yemon Choi Sep 2 '12 at 6:03

I have studied both the "elementary" and "analytic function theoretic" proofs of PNT and I find that the analytic version is more natural and appealing whereas the so called "elementary" version looks more crafted specifically to avoid complex analysis.

I can illustrate my point by a very simple example. While studying calculus I learned the technique of integration by parts and using this one could evaluate the integral $\displaystyle \int e^{ax}\sin bx\\,dx$.

But a simple trick using complex numbers would make the evaluation of the integral so easy and at the same time so marvelous:

$\displaystyle A = \int\\,e^{ax}\cos bx\\,dx, \\,\\, B = \int\\,e^{ax}\sin bx\\,dx$

so that $\displaystyle A + iB = \int\\,e^{(a + ib)x}\\,dx = \frac{e^{(a + ib)x}}{a + ib}$

The answer follows by equating real and imaginary parts. This uses only the algebra of complex numbers.

The Cauchy's integral theorem which lies at the heart of all complex analysis is way more powerful than it seems (on a first sight) and can make various proofs much more simple and natural than their elementary counterparts.

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This doesn't seem to be an answer to the question. –  Gerry Myerson Sep 2 '12 at 5:30

Taking complex analysis from number theorists is like taking away their lives. So called "elementary is just relativly simple and direct.

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I don't think so. The complex variable proofs of the Prime Number Theorem are generally held to be simpler than the elementary proofs, and I'm not sure the word "direct" can be defined sharply enough to get a consensus on which approach is the more direct. –  Gerry Myerson Aug 24 '10 at 6:34

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