# Representation of an integer as the sum of a prime in arithmetic progression and a square-free integer

## TL;DR

This paper develops an abstract circle method to estimate the number of representations of large integers as the sum of a prime in an arithmetic progression and a square-free integer, uniformly for small moduli.

## Contribution

It introduces a novel application of the local model approach to the circle method for problems involving primes and square-free integers.

## Key findings

- Provides an estimate for the weighted count of such representations
- Achieves uniformity for small moduli q
- Extends circle method techniques to new additive problems

## Abstract

Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramar\'e and may be viewed as an abstract circle method.