How many ways can a positive integer $n$ be written as a sum of squares? For a fixed elliptic curve $E$ over $\mathbb{Q}$, how many primes $p$ will have a maximal number of points (with respect to the Hasse bound) on the reduced curve over $\mathbb{F}p$? For which subsets $A\subset \mathbb{N}$ and real numbers $\alpha$ does the sequence $(\alpha x){x\in A}$ modulo 1 exhibit “random” behavior?

On the surface, these problems do not seem related. However, they are all connected by the methods used in the proof. At their core, these are all counting problems. In this talk, we will discuss various tools from complex and Fourier analysis that can be employed to answer the questions above, as well as many other problems in analytic number theory and combinatorics.