A note on the least totient of a residue class

Let $q$ be a large prime number, $a$ be any integer, $ϵ$ be a fixed small positive quantity. Friedlander and Shparlinksi \cite{FSh} have shown that there exists a positive integer $n\ll q^{5\mathrm{/}2+ϵ}$ such that $\varphi (n)$ falls into the residue class $a(\mathrm{m}\mathrm{o}\mathrm{d}q)\mathrm{.}$ Here, $\varphi (n)$ denotes Euler’s function. In the present paper we improve this bound to $n\ll q^{2+ϵ}\mathrm{.}$