Least Totients in Arithmetic Progressions


Let N(a, m) be the least integer n (if exists) such that φ(n) ≡ a (mod m). Friedlander and Shparlinski proved that for any ε > 0 there exists A = A(ε) > 0 such that for any positive integer m which has no prime divisors p < (log m) and any integer a with gcd(a,m) = 1, we have the bound N(a,m) ¿ m. In the present paper we improve this bound to N(a,m) ¿ m.


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