Backstory: Frenkel's Metaphors and Waitressing for Godot.
In a recent vulgarized presentation of the Langlands program,
Edward Frenkel implied that number theory and harmonic
analysis were, before Langlands came along, quite unrelated.
This is false.
"If we think of different fields of mathematics as continents,
then number theory would be like North America and
harmonic analysis like Europe."
— Edward Frenkel, Love and Math , 2013
For a discussion of pre-Langlands connections between
these "continents," see…
"Fourier Analysis in Number Theory, my senior thesis, under the advisory of Patrick Gallagher.
This thesis contains no original research, but is instead a compilation of results from analytic
number theory that involve Fourier analysis. These include quadratic reciprocity (one of 200+
published proofs), Dirichlet's theorem on primes in arithmetic progression, and Weyl's criterion.
There is also a function field analogue of Fermat's Last Theorem. The presentation of the
material is completely self-contained."
— Shanshan Ding, University of Pennsylvania graduate student