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I read on Wikipedia that an elementary proof in number theory means a proof which does not use complex analysis.

From what I recall reading, in Hardy's time, proofs avoiding complex analysis were preferred.

I would like to ask these two questions, with the stress on the second one:

  • What were the reasons that proofs without complex analysis were preferred?
  • Is this distinction still important today?

This question arose from a discussion on meta about this post.

I have seen this related post, but I'd say it is not the same question: Elementary proof of the Prime Number Theorem - Need?. That question is asking more about whether finding an elementary proof had some important consequences for using similar method in other areas. (But Matt E's answer posted there also deals with the motivation for the search for an elementary proof. So it can also be considered an answer to the first bullet above.)

This MathOverflow post is also interesting in this context: Complex and Elementary Proofs in Number Theory. It discusses where number-theoretic results based on complex analysis can also be shown without using complex-analytic methods.

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  • $\begingroup$ Of course I might have misjudged to which extent the two questions are really different. (Deciding whether two questions are exact duplicates is always a bit subjective. I agree that they are closely related.) So if you think that this is duplicate, definitely go ahead and say so in comments/vote to close as a duplicate. $\endgroup$ – Martin Sleziak Jan 8 '15 at 8:22
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    $\begingroup$ Another related MO post $\endgroup$ – Zubin Mukerjee Jan 8 '15 at 8:23
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In the Elementary proof of the prime number theorem - Need? I address a bit about this, but I agree your question is distinct.


What were the reasons that proofs without complex analysis were preferred?

As I have heard from Jeffrey Vaaler, and others who were around closer to the time in question, there was something of a belief that a proof without analysis would somehow illuminate something deep about the complexity of the results involved. Evading the use of complex analysis would mean that it was perhaps "less difficult than we thought," or at least this was the view at the time. This reflected the desire to understand the exact nature of the theorem, was it fundamentally related to arithmetic or was the excursion into the Fourier side of things truly necessary, and if so: why? This, to me at least, seems quite natural for a mathematician, as we seek not just to know results but to understand them correctly. Lagrange's theorem is really a statement about the homogeneous nature of groups, something that gives motivation and intuition for things like topological groups, even though the former is just a statement about divisibility of orders. There are, of course many more examples, but as you also have the response in the other topic, I'll curtail further exposition on my part for this bullet point.

Is this distinction still important today?

I haven't felt that at all in the conferences, papers, et cetera that I've read in today's number theory environment. It seems that was a popular sentiment at the time, but--especially following the lack of really interesting new results that could be proved with the Erdös-Selberg proof--that has died down significantly to where it is no longer detectable, even among those who were immediate successors of mathematicians like Hardy.

A lot like proofs that don't use the axiom of choice, it's become less of a community preoccupation over time. In the same way that we all somehow "get used to" new results in mathematics and become less wary as the details of the proof get easier to understand with repetition of reading, things like the axiom of choice have become routine. I think this is an exact analogue with the elementary proof, in despite of all the solid foundation, the culture at the time simply valued elementary techniques more highly. As amazingly useful as harmonic analysis is today, it wasn't all the way as developed as it is now. The Wiener-Ikehara theorem was only published in 1931, and Tate's thesis which brought things to a very interesting point with more abstract harmonic analysis rather than strictly on Euclidean space came in 1950, and Rudin's text on Fourier analysis on groups was in 1962, which was after Hardy's time.

In short more modern approaches were developed, tried by fire, and found to be superior to many classical proofs and techniques. In a sort of academic survival of the fittest, the elementary approach was found unfavorable by the large majority after it was unable to deliver what was hoped, and other techniques have since supplanted them as the more "in vogue" ways to approach problems. There are certainly places where excellent new mathematics comes out of elementary approaches, UIUC for one has many excellent number theorists who are skilled at such things. I do not think the number theory community at-large looks with greater favor upon such approaches like it did in Hardy's day.

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I don't think elementary proofs are necessarily preferred. But mathematicians are always keen on different methods to solve the same problem because all of them usually offer different insights. And especially in the case of primes "elementary" methods are interesting since primes are such an elementary construct.

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I think the "elementary"/"non-elementary" distinction is purely an anachronism dating from the 18th century and prior, when there was still a lot of suspicion about complex numbers and analytic methods deriving from them. A good indicator of this is the term "imaginary numbers" for elements of the Argand plane-as the late great George Carlin once said, language always gives away the true intent behind words and phrases. In any event, there's no inherent difficulty in complex methods proofs in analytic number theory compared with other methods, which is why the phrase really appears rather silly now.

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    $\begingroup$ It wasn't silly to Erdos or Selberg. Have we evolved that quickly? $\endgroup$ – daniel Jan 8 '15 at 9:07
  • $\begingroup$ @daniel Erdős and Selberg were working on this more than 65 years ago. That's not evidence of quick evolution. $\endgroup$ – David Richerby Jan 8 '15 at 12:57
  • $\begingroup$ @daniel It wasn't silly to them because like all good research mathematicians, Erdos and Selberg were looking to demonstrate a way of doing the impossible. Most mathematicians at that point had begun to doubt a non-complex number based proof of the Prime Number Theorem was possible.I don't really think it had anything to do with an "elementary" method being preferable or easier to the complex version-it was just more novel and creative. $\endgroup$ – Mathemagician1234 Jan 15 '15 at 23:06
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    $\begingroup$ @Mathemagician1234: You have posted some really nice answers but this one seems anhistorical. Adam Hughes' observation that the distinction became "less of a community preoocupation over time" is certainly correct. But Selberg didn't get a Fields medal for wrestling with an "anachronism dating from the 18th century." When Ingham reviewed Erdos' and Selberg's papers he used the term transcendental to refer to complex methods. There was a feeling that the power of complex methods transcended elementary methods. That sense has faded but I don't think it was ever seen as silly. $\endgroup$ – daniel Jan 16 '15 at 7:06
  • $\begingroup$ @daniel Selberg shared a Fields medal for co-discovering a very novel means of proving an important result using methods most mathematicians at the time thought were insufficient. He dispelled as false a perception that the majority of mathematicians had. Also, when I said the distinction between complex and elementary methods in number theory was silly, I was referring to the original motivation for the distinction in the first place-namely, the initial distrust in mathematics for complex numbers and functions. That was long over by the time Erdos and Selberg published their proof. $\endgroup$ – Mathemagician1234 Jan 18 '15 at 17:37

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