Smallest totient in a residue class

Abstract

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers $(m,a)$ such that $m$ is odd, there exists $n\le m^{2+o(1)}$ such that $\ddot{I}†(n)\equiv a\mathrm{mod}m$.

To appear in Bulletin of London Mathematical Society