Analytic Number Theory without Pigeonhole Principle 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. 
 A: 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.
