On two conjectures concerning squarefree numbers in arithmetic progressions

TL;DR

This paper advances the understanding of how squarefree numbers distribute in arithmetic progressions by establishing new upper bounds on error terms, leveraging recent exponential sum estimates to make progress on longstanding conjectures.

Contribution

It provides improved upper bounds for the distribution error of squarefree numbers in arithmetic progressions, utilizing recent exponential sum estimates, thus advancing towards two famous conjectures.

Findings

Improved upper bounds for error terms in distribution of squarefree numbers

Progress towards two well-known conjectures in number theory

Enhanced estimates using Bourgain-Garaev and Bourgain-Fouvry-Kowalski-Michel results

Abstract

We prove upper bounds for the error term of the distribution of squarefree numbers up to $X$ in arithmetic progressions modulo $q$ making progress towards two well-known conjectures concerning this distribution and improving upon earlier results by Hooley. We make use of recent estimates for short exponential sums by Bourgain-Garaev and for exponential sums twisted by the M\"obius function by Bourgain and Fouvry-Kowalski-Michel.