Squarefree Integers in Arithmetic Progressions to Smooth Moduli

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