Ok so i got my PhD in connections between string theory and number theory. I think my favorite example of this connection is the fact that black hole entropies are given by the Fourier coefficients of modular forms. This is particularly clear in the context of so-called small black holes, where literally the number of states in a black hole of mass $n$ is given by the $n$-th Fourier coefficient of 1/Delta(tau). where Delta is the weight-12 cusp form. In fact, the Fourier coefficients of this modular form admit an exact series expansion known as a Rademacher series, which takes the form of a sum over elements of (a quotient of) SL_2(Z). The *exact same series* appears in the evaluation of the gravitational path integral which is supposed to compute this entropy, so we have a highly nontrivial matching between the number theory result and actual string theory.
I can go on for hours about other connections between string theory and number theory (or, ya know, write a thesis about it :p), but I think this is the most striking demonstration.
