Square values of Euler's function

Euler's function, ubiquitous in number theory, has been studied as an arithmetic function for a long time.  We know it's average to $x$ fairly well, we know it's distribution, and we know a lot about the frequency of integers (called totients) which are values of Euler's function.  This talk concerns square totients.  It is perhaps surprising that up to $x$ there are many more integers that Euler's function maps to a square than there are squares themselves.  However, are most squares totients?  Surely the answer should be ``no'' and in this paper we prove this.  Perhaps there is an easier path, but our proof is surprisingly difficult.  (Joint work with Paul Pollack)