Squarefree Integers in Arithmetic Progressions to Smooth Moduli

TL;DR

This paper proves an asymptotic formula with power-saving error terms for counting squarefree integers in arithmetic progressions with smooth, squarefree moduli, improving previous bounds and extending distribution results.

Contribution

It introduces new bounds for squarefree integers in arithmetic progressions with smooth moduli, surpassing previous limitations and establishing a higher level of distribution.

Findings

Improves the level of distribution for squarefree integers in arithmetic progressions.

Breaks the $X^{3/4}$-barrier for a density 1 set of smooth moduli.

Uses advanced exponential sum estimates and cohomological methods.

Abstract

Let $\epsilon > 0$ be sufficiently small and let $0 < \eta < 1/522$. We show that if $X$ is large enough in terms of $\epsilon$ then for any squarefree integer $q \leq X^{196/261-\epsilon}$ that is $X^{\eta}$-smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod{q}$, with $(a,q) = 1$. In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096...$ was replaced by $25/36 = 0.69\bar{4}$. This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$-barrier for a density 1 set of $X^{\eta}$-smooth moduli $q$ (without the squarefree condition). Our proof appeals to the $q$-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using $p$-adic methods.