A note on the least totient of a residue class

الأصل · EN

Let q be a large prime number, a be any integer, ε be a fixed small positive quantity. Friedlander and Shparlinksi FSh have shown that there exists a positive integer n≪ q⁵/²⁺ε such that ϕ(n) falls into the residue class a q. Here, ϕ(n) denotes Euler's function. In the present paper we improve this bound to n≪ q²⁺ε.