Smallest totient in a residue class

TL;DR

This work proves a totient analogue of Linnik's theorem in arithmetic progressions: for any coprime pair $(m,a)$ with $m$ odd, there exists $n \le m^{2+o(1)}$ such that $\varphi(n) \equiv a \pmod m$. The authors construct $n$ as a product of three primes (up to a power of 4 if needed) with primes chosen from carefully defined intervals and count solutions via Dirichlet character sums, splitting the analysis into small and large conductor cases. They derive precise asymptotics for the small-conductor sums and bound the large-conductor contribution using advanced estimates for character sums (including Bombieri–Vinogradov-type results and Hölder’s inequality). The result yields a uniform bound for $N(a,m)$, advancing our understanding of how totients populate residue classes and paralleling Linnik-type results for primes.

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 $\varphi(n)\equiv a\,\mathrm{mod}\,{m}$.

Theorems & Definitions (10)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Remark 4.1
• Lemma 4.2
• proof