An area where analysis (especially the Fourier kind) is used quite heavily is in the study of so-called "Beurling-Selberg maximal functions," which have applications to several areas in number theory, such as counting lattice points and studying the behavior of zeta- and $L$-functions. The basic idea is that there are some naturally occurring functions, such as characteristic functions of intervals, balls, etc., which obviously don't have nice analytic properties, and it can be very fruitful to find functions which majorize and minorize your "bad" function that have "nice" properties with respect to their fourier transforms.

The paper entitled

A survey on Beurling-Selberg majorants and some consequences of the Riemann hypothesis

by Emanuel Carneiro seems like it would be a good starting point, but I can't seem to find a full text online. Carneiro and others have written quite a few papers on this topic, such as this one:

https://www.ndsu.edu/pubweb/~littmann/research/Gaussian_2_4.pdf

by Carneiro, Littmann, and Vaaler, which I would describe as mostly analysis, with some discussion of applications to number theory. You might find this stuff interesting.