Mertens and Möbius function
Sorry, only two links today: check out the Mertens and Möbius functions. The former is built upon the latter. Its structure is surprising and almost remeniscent of random behaviour, but completely random it is not. I just find it mesmerizing.
The Mertens functions is closely related to the world famous Riemann zeta function. This latter function is conjectured to have a certain property, but while few people doubt the veracity of the statement, nobody in over a century has been able to find a proof.