Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Alexander P. Mangerel

Forum of Mathematics, Sigma, Tome 9 (2021)

Résumé

Let $\varepsilon> 0$ be sufficiently small and let $0 \eta 1/522$. We show that if X is large enough in terms of $\varepsilon $, then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ 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\ldots $ was replaced by $25/36 = 0.69\overline {4}$. 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.

Détail

DOI : 10.1017/fms.2021.67