Let ε>0 be sufficiently small and let 0<η<1/522 . We show that if X is large enough in terms of ε , then for any squarefree integer q≤X196/261−ε that is Xη -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression a(modq) , 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 ¯ 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 X3/4 -barrier for a density 1 set of Xη -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.
Mangerel, A. P. (2021). Squarefree Integers in Arithmetic Progressions to Smooth Moduli. Forum of Mathematics, Sigma, 9, Article e72. https://doi.org/10.1017/fms.2021.67