Squarefree Integers in Arithmetic Progressions to Smooth Moduli release_5qrwgtqrdzf3del2wexspdvfkq

Abstract

<jats:title>Abstract</jats:title> Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline1.png" /> <jats:tex-math> $\varepsilon&gt; 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be sufficiently small and let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline2.png" /> <jats:tex-math> $0 &lt; \eta &lt; 1/522$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that if <jats:italic>X</jats:italic> is large enough in terms of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline3.png" /> <jats:tex-math> $\varepsilon $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then for any squarefree integer <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline4.png" /> <jats:tex-math> $q \leq X^{196/261-\varepsilon }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that is <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline5.png" /> <jats:tex-math> $X^{\eta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline6.png" /> <jats:tex-math> $a \pmod {q}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline7.png" /> <jats:tex-math> $(a,q) = 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline8.png" /> <jats:tex-math> $196/261 = 0.75096\ldots $ </jats:tex-math> </jats:alternatives> </jats:inline-formula> was replaced by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline9.png" /> <jats:tex-math> $25/36 = 0.69\overline {4}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline10.png" /> <jats:tex-math> $X^{3/4}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-barrier for a density 1 set of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509421000670_inline11.png" /> <jats:tex-math> $X^{\eta }$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-smooth moduli <jats:italic>q</jats:italic> (without the squarefree condition). Our proof appeals to the <jats:italic>q</jats:italic>-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 <jats:italic>p</jats:italic>-adic methods.
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