On the distribution of squarefree integers in arithmetic progressions

TL;DR

This paper studies the distribution of squarefree integers in arithmetic progressions, providing improved bounds on the variance of their count, which advances understanding of their distribution in number theory.

Contribution

It introduces a new upper bound for the variance of squarefree integers in arithmetic progressions, improving upon previous results by Blomer.

Findings

Established a tighter upper bound for variance

Enhanced understanding of squarefree integers distribution

Improved upon previous theoretical bounds

Abstract

We investigate the error term of the asymptotic formula for the number of squarefree integers up to some bound, and lying in some arithmetic progression a (mod q). In particular, we prove an upper bound for its variance as a varies over $(\mathbb{Z}/q\mathbb{Z})^{\times}$ which considerably improves upon earlier work of Blomer.