# On products of primes and square-free integers in arithmetic progressions

## TL;DR

This paper derives an asymptotic formula for counting representations of residue classes as products of a prime and a square-free integer, advancing understanding of prime-related structures in arithmetic progressions.

## Contribution

It provides a new asymptotic formula for representing residue classes as products of primes and square-free integers, addressing a relaxed form of a conjecture by Erdős, Odlyzko, and Sárközy.

## Key findings

- Established an asymptotic count for such representations.
- Connected the problem to a relaxed version of a classical conjecture.
- Enhanced understanding of prime and square-free integer distributions in residue classes.

## Abstract

We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erd\"os, Odlyzko, and S\'ark\"ozy.