Squarefree Integers in Arithmetic Progressions to Smooth Moduli
- Mangerel, Alexander P.
DOIAbstract
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 appea...
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