In this talk we shall take an eclectic tour through three centuries since the inception of this function by Euler in 1729. The function has been the object of intense study by every great mathematician of the 19th century and continues to tantalize the modern mathematicians.

We shall dwell upon some of the more important identities, their modern ramifications and some new proofs of old results. We shall also discuss some recent developments that have occured in the last decade.