As far as I know, in analytic number theory, harmonic analysis appears often. The thing is that I would see the proof of some results where they use harmonic analysis, and I can follow the argument of the proof and it makes sense, but I have no intuition behind why one would consider using harmonic analysis there (other than that using it works…).
For example, maybe in a proof one has to estimate a sum of the form $\sum f(n)$ and so they would take the Fourier transform and use Poisson summation formula or something and it works. I would understand the proof, but I just have no idea why it was the "right" thing to do or why it was a good thing to try (other than of course that it worked out).
I know my question is rather vague, but I would appreciate some explanations if possible! Also I would try to modify the question in a better way if anyone has any suggestion. Thank you very much!
Best Answer
Harmonic analysis is the theory of representations of locally compact abelian groups. The integers, and the integers mod $n$, are such groups. For problems dealing with functions on these groups that are related to their group structure, harmonic analysis is a natural tool.