I believe a typical key result in number theory whose proof relies (not only, but critically) on functional analysis is the [Selberg trace formula][1] and its variants. More generally, the theory of automorphic forms heavily depends on spectral theory, as one can see [in this post in Terence Tao's blog][2].


  [1]: https://en.wikipedia.org/wiki/Selberg_trace_formula
  [2]: https://terrytao.wordpress.com/2015/08/15/a-wave-equation-approach-to-automorphic-forms-in-analytic-number-theory/