As far as I know in analytic number theory, harmonic analysis appears often. The thing is 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!