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Keywords:
$k$-free number; exponential sum; Beatty sequence
$k$-free number; exponential sum; Beatty sequence
Summary:
We consider $k$-free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $\alpha >1$ of finite type $\tau <\infty $ and any constant $\varepsilon >0$, we can show that $$ \sum _{ 1\leq n\leq x \atop [\alpha n+\beta ]\in \mathcal {Q}_{k}} 1- \frac {x}{ \zeta (k)} \ll x^{k/(2k-1)+\varepsilon }+x^{1-1/(\tau +1)+\varepsilon }, $$ where $\mathcal {Q}_{k}$ is the set of positive $k$-free integers and the implied constant depends only on $\alpha ,$ $\varepsilon ,$ $k$ and $\beta .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $$ \sum _{1\leq h\leq H}\sum _{ 1\leq n\leq x \atop n\in \mathcal {Q}_{k}}e(\vartheta hn). $$
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