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I don't know if this is an appropriate question for this website, but I will try my luck.

I am an undergraduate student, and recently I became interested in analytic number theory. When I started reading introductory material in the subject, I got the idea that most "structures" and most ideas in proofs are heavily based on combinatorial-like ideas. For example, the Pigeonhole Principle is used extensively.

Of course some people would find this very obvious since the subject is concerned primarily with things like "counting the number of primes less than a given magnitude". However, I would still argue that things could have been very different and combinatorial techniques could have been ineffective.

So my question is: First of all, in general, could techniques from combinatorics be replaced by other methods in order to achieve the same results we have today.

Secondly, is there a "bigger picture"? In other words, do we know of a higher mathematical structure which explains this intersection between analytic number theory and combinatorics.

Please excuse me if you find that questions are badly formulated. After all, English is not my mother language.

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    Honestly, I don't quite understand the point of this kind of questions: "do this without that". (Of course, especially this concerns the axiom of choice, without which pretty much nothing works in math.) Why not treat math a a single subject where everything is intertwined? (That's why I'm not downvoting, as your second question fits into this framework :) – Alex Degtyarev Jul 22 '15 at 21:51
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    I love the axiom of choice, but I have to object to ". . . without which pretty much nothing works in math": plenty of things don't involve choice at all, and many things work better without it and with determinacy (or similar) instead. – Noah Schweber Jul 22 '15 at 21:53
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    I think your impression that analytic number theory is heavily based around combinatorial ideas is incorrect. While I am far from an expert in the subject, my impression is that most work in the subject uses serious analytic tools that are not really combinatorial, e.g. Fourier analysis as in the circle method and complex analysis. – Andy Putman Jul 22 '15 at 22:00
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    @AlexDegtyarev: Isn't "do this without that" the whole idea of reverse mathematics? I think this particular question is too broad for this site, but in general it's a legitimate sort of question to ask. – Nate Eldredge Jul 22 '15 at 22:04
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    Maybe these go some way towards an answer: http://mathoverflow.net/questions/36405/complex-and-elementary-proofs-in-number-theory http://mathoverflow.net/questions/61632/what-is-the-high-concept-explanation-on-why-real-numbers-are-useful-in-number-th – Dan Piponi Jul 22 '15 at 22:12
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    The question that is specifically asked here is quite unclear to me, but nevertheless I believe that reading Elkies's "What is analytic number theory?" might help to clear up the underlying question. –  Jul 22 '15 at 23:06
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    I agree with @AndyPutman and dispute the claim made in the 2nd paragraph of this question, which seems to me to be based on (understandable) inexperience. – Yemon Choi Jul 22 '15 at 23:10
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    I'm voting to close this question as off-topic because I think it is based on a false premise – Yemon Choi Jul 22 '15 at 23:11
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    @AlexDegtyarev One of the most important math questions in history is "Can one do geometry without the parallel postulate?" Not all questions of this form lead anywhere useful, but many do. – Paul Siegel Jul 22 '15 at 23:52
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    The Pigeonhole Principle is logically equivalent to Mathematical Induction. Induction is the distinctive feature of the natural numbers. So it shouldn't be surprising that the Pigeonhole Principle is used extensively in any number-theoretic pursuit. – Gerry Myerson Jul 23 '15 at 00:01
  • I'm not a number theorist, but from what I remember of my analytic numbery I can sympathize with this question; lots of proofs proceed by constructing more than $n$ conjugacy classes modulo $n$. Maybe others will like the question better if you can focus on a particularly striking example of this phenomenon. – Paul Siegel Jul 23 '15 at 00:02
  • G. Myerson above is correct. That is the main reason one cannot do without it. I have seen (failed) attempts to get rid of it because it's somehow nonconstructive but those were not persuasive. – Gottfried William Jul 24 '15 at 03:48

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To my perception, the reason "elementary" number theory (which includes some "analytic" number theory, which, ironically, can be manifest in ways which involves no analysis whatsoever) seems "combinatorial", is that a naive sense of "combinatorial" is a fairly superficial one of "involving no structure or big prerequisites".

So, sure, elementary number theory can be done, if one insists, without too many prerequisites. That doesn't really mean that it's "combinatorial", unless the latter is taken as synonym for "without prerequisites". I do suspect that this slippery slope is what leads many people to ask questions based on an over-interpretation of the facts... and mis-labeling.

The specific example of "pigeon-hole principle" is misleading in at least two ways. One, if this is "combinatorial", then everything is combinatorial. Second, there are certainly very-non-finitistic versions of this, very non-elementary, so it's hard to really claim that this is "combinatorial" in any sense that usefully distinguishes it for "other mathematics".

A second way that "a higher something" may "suggest" an interaction between elementary number theory and "combinatorics" is simply that many more sophisticated structures can be denatured to an extent, to "combinatorial/finitistic/formulaic" assertions, which can sometimes be a sufficient skeletal causality to prove some number-theoretic facts. But this is a special case of the general principle that anything can be denatured (selectively or accidentally or...) to look "combinatorial" but still manage to minimally succeed.

E.g., Fermat's little theorem can be construed as something "combinatorial" about binomial coefficients ... but it also can be viewed as an instantaneous consequence of Lagrange's theorem in group theory.

At a different extreme, if someone wants to claim that "sieving" (e.g., see recent work of Zhang, Maynard, Tao, et al) is "combinatorial", well, ... :)

So, quite seriously, I think that perception of number theory as "combinatorial" is misguided, perhaps misguided semantically, unless one makes the word "combinatorial" be so broad as to be useless.

paul garrett
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  • Well, I agree that in some sense all of mathematics is combinatorial. However it seems to me (of course I could be completely wrong) that some areas are more so than others (finite group theory vs differential geometry, for example). As for analytic number theory, it seems to me that there is an intersection between the discrete and the continuous, therefore, I think that there is a "higher structure" which explains this intersection. –  Jul 22 '15 at 23:05
  • It is still not clear to me what you mean by "analytic number theory". Please clarify? There is much potential ambiguity in that label. E.g., do you mean Iwasawa-Tate theory of zeta functions? Jacquet-Langlands? Sieves? The Zhang-Maynard-Tao business? Subtle things about moment estimates? Subconvexity? "The thing is", most of these are not really "combinatorial" or "discrete" in any operational sense, so there's some element of mis-reference, to my mind. Please clarify? – paul garrett Jul 22 '15 at 23:17
  • By analytic number theory, I mean sieve theory especially as you said, all the Zhang-Maynard-Tao business. –  Jul 22 '15 at 23:24
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    When I describe a result as "combinatorial" I usually mean that it involves delicately counting something (as opposed to computing, comparing, axiomatizing...) I think the OP is asking: "Can you do number theory without counting?" It doesn't seem intuitively likely, but who knows? – Paul Siegel Jul 23 '15 at 00:10