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.

withoutit and with determinacy (or similar) instead. $\endgroup$ – Noah Schweber Jul 22 '15 at 21:53