Least totients in arithmetic progressions

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