The Ramanujan sum c_q(n) is the sum of the nth powers of the primitive qth roots of unity. Ramanujan sums satisfy "orthogonality relations," which raises the possibility of writing a general arithmetic function as an infinite sum of Ramanujan sums with complex coefficients similar to a Fourier series. In this talk, I will prove some basic properties of Ramanujan sums, including the orthogonality relations. We'll look at some examples of Ramanujan expansions of arithmetic functions found by Ramanujan and Hardy. Finally, I'll show how one of Hardy's examples, combined with a formula analogous to the Wiener-Khinchin Theorem in Fourier analysis, "almost" proves the twin prime conjecture. Speaker(s): Corey Everlove (UM)