Every integer can be written as a square plus a squarefree

TL;DR

This paper proves every integer can be expressed as a sum of a perfect square and a squarefree number, providing an asymptotic count and new bounds related to the divisor function.

Contribution

It introduces a novel representation of integers as a sum of a square and a squarefree number, with asymptotic formulas and explicit divisor function bounds.

Findings

Every integer can be written as a square plus a squarefree number.

Established an asymptotic formula for the number of such representations.

Derived a new explicit upper bound for the divisor function.

Abstract

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result is deeply related with the divisor function. In the course of our study we get an independent result about it. Concretely we are able to deduce a new upper bound for the divisor function valid for any integer and fully explicit.