# Product of an integer free of small prime factors and prime in arithmetic progression

## TL;DR

This paper investigates the representation of reduced residue classes as products of primes and integers free of small prime factors, providing new estimates and progress on longstanding conjectures, with some results conditional on GRH.

## Contribution

It introduces new estimates for such representations and advances the understanding of a conjecture by Erdös, Odlyzko, and Sárközy, under GRH assumptions.

## Key findings

- Conditional estimates for the number of representations
- Progress on Erdös-Odlyzko-Sárközy conjecture
- Results depend on the Generalised Riemann hypothesis

## Abstract

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erd\"os, Odlyzko and S\'ark\"ozy.