TITLE: Spinoffs of analytic number theory
QUESTION [11 upvotes]: What are some techniques and theorems of analytic number theory that have proved useful outside of number theory?

REPLY [2 votes]: The circle method was an inspiration behind the solution of the toroidal semiqueens problem by Eberhard, Manners and Mrazović ("Additive triples of bijections, or the toroidal semiqueens problem", JEMS 21 (2019), no. 2, 441–463).
The problem is equivalent to counting permutations $\pi_1,\pi_2$ of $\mathbb{Z}/n\mathbb{Z}$ whose pointwise sum is also a permutation. The proof adapts the circle method to the group $(\mathbb{Z}/n\mathbb{Z})^n$.