Smallest totient in a residue class

TL;DR

This paper establishes a new result in number theory, showing that for any coprime pair with an odd modulus, there exists a number with a small totient satisfying a specific modular condition, extending Linnik's theorem analogously.

Contribution

It provides a totient analogue of Linnik's theorem in arithmetic progressions, demonstrating the existence of small totients in a given residue class for odd moduli.

Findings

Existence of n ≤ m^{2+o(1)} with φ(n) ≡ a mod m for coprime (m,a) with m odd.

Extension of Linnik's theorem to totients in arithmetic progressions.

New bounds on the size of integers with prescribed totient residue.

Abstract

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers $(m,a)$ such that $m$ is odd, there exists $n\le m^{2+o(1)}$ such that $\varphi(n)\equiv a\,\mathrm{mod}\,{m}$.