A note on the least totient of a residue class

TL;DR

This paper improves the upper bound on the size of integers with Euler's totient in a given residue class modulo a large prime, reducing it from roughly q^{5/2} to q^{2} with a small epsilon.

Contribution

The authors improve the bound on the least integer with a given totient residue class modulo a prime from q^{5/2+ε} to q^{2+ε}.

Findings

Bound on n is improved to n ≪ q^{2+ε}.

The result applies to large prime moduli q.

Enhances understanding of Euler's totient distribution in residue classes.

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 $n\ll q^{2+\epsilon}.$