Residue classes free of values of Euler's function

Abstract: We characterize which residue classes contain infinitely many totients (values of Euler's function) and which do not. We show that the union of all residue classes that are totient-free has asymptotic density 3/4, that is, almost all numbers that are 2 mod 4 are in a residue class that is totient-free. In the other direction, we show the existence of a positive density of odd numbers m, such that for any $s\ge0$ and any even number $a$, the residue class $a\pmod{2^sm}$ contains infinitely many totients.