A note on primes in certain residue classes

TL;DR

This paper establishes a lower bound on the asymptotic density of primes avoiding certain residue classes, extending classical results about integers to primes with a similar density estimate.

Contribution

It proves a new density result for primes in specific residue classes, generalizing previous integer-based theorems to the prime setting.

Findings

The set of primes avoiding certain residue classes has a density at least the product of (1 - 1/φ(a_i)).

The result parallels classical density theorems for integers by Heilbronn and Rohrbach.

Provides a quantitative measure of how primes distribute among residue classes.

Abstract

Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{\varphi(a_i)}\right)$, where $\varphi$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.