On $k$-free numbers over Beatty sequences

Zhang, Wei. "On $k$-free numbers over Beatty sequences." Czechoslovak Mathematical Journal 73.3 (2023): 839-847. <http://eudml.org/doc/299124>.

@article{Zhang2023,
abstract = {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 \[ \sum \_\{ 1\le n\le x \atop [\alpha n+\beta ]\in \mathcal \{Q\}\_\{k\}\} 1- \frac\{x\}\{ \zeta (k)\} \ll x^\{k/(2k-1)+\varepsilon \}+x^\{1-1/(\tau +1)+\varepsilon \}, \] 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 \[ \sum \_\{1\le h\le H\}\sum \_\{ 1\le n\le x \atop n\in \mathcal \{Q\}\_\{k\}\}e(\vartheta hn). \]},
author = {Zhang, Wei},
journal = {Czechoslovak Mathematical Journal},
keywords = {$k$-free number; exponential sum; Beatty sequence},
language = {eng},
number = {3},
pages = {839-847},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On $k$-free numbers over Beatty sequences},
url = {http://eudml.org/doc/299124},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Zhang, Wei
TI - On $k$-free numbers over Beatty sequences
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 839
EP - 847
AB - 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 \[ \sum _{ 1\le n\le x \atop [\alpha n+\beta ]\in \mathcal {Q}_{k}} 1- \frac{x}{ \zeta (k)} \ll x^{k/(2k-1)+\varepsilon }+x^{1-1/(\tau +1)+\varepsilon }, \] 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 \[ \sum _{1\le h\le H}\sum _{ 1\le n\le x \atop n\in \mathcal {Q}_{k}}e(\vartheta hn). \]
LA - eng
KW - $k$-free number; exponential sum; Beatty sequence
UR - http://eudml.org/doc/299124
ER -