A note on the least totient of a residue class

Abstract

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