Least totients in arithmetic progressions

Abstract:

Let $N(a,m)$ be the least integer $n$ (if it exists) such that $\varphi (n)\equiv a\pmod m$. Friedlander and Shparlinski proved that for any $\varepsilon >0$ there exists $A=A(\varepsilon )>0$ such that for any positive integer $m$ which has no prime divisors $p<(\log m)^A$ and any integer $a$ with $\gcd (a,m)=1,$ we have the bound $N(a,m)\ll m^{3+\varepsilon }.$ In the present paper we improve this bound to $N(a,m)\ll m^{2+\varepsilon }.$