On the r -free values of the polynomial x 2 + y 2 + z 2 + k

Gongrui Chen; Wenxiao Wang

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 3, page 955-969
  • ISSN: 0011-4642

Abstract

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Let k be a fixed integer. We study the asymptotic formula of R ( H , r , k ) , which is the number of positive integer solutions 1 ≤ x , y , z ≤ H such that the polynomial x 2 + y 2 + z 2 + k is r -free. We obtained the asymptotic formula of R ( H , r , k ) for all r ≥ 2 . Our result is new even in the case r = 2 . We proved that R ( H , 2 , k ) = c k H 3 + O ( H 9 / 4 + ε ) , where c k > 0 is a constant depending on k . This improves upon the error term O ( H 7 / 3 + ε ) obtained by G.-L. Zhou, Y. Ding (2022).

How to cite

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Chen, Gongrui, and Wang, Wenxiao. "On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$." Czechoslovak Mathematical Journal 73.3 (2023): 955-969. <http://eudml.org/doc/299125>.

@article{Chen2023,
abstract = {Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x, y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)= c_kH^3 +O(H^\{9/4+\varepsilon \})$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O(H^\{7/3+\varepsilon \})$ obtained by G.-L. Zhou, Y. Ding (2022).},
author = {Chen, Gongrui, Wang, Wenxiao},
journal = {Czechoslovak Mathematical Journal},
keywords = {square-free; Salié sum; asymptotic formula},
language = {eng},
number = {3},
pages = {955-969},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$},
url = {http://eudml.org/doc/299125},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Chen, Gongrui
AU - Wang, Wenxiao
TI - On the $r$-free values of the polynomial $x^2 + y^2 + z^2 +k$
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 955
EP - 969
AB - Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x, y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)= c_kH^3 +O(H^{9/4+\varepsilon })$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O(H^{7/3+\varepsilon })$ obtained by G.-L. Zhou, Y. Ding (2022).
LA - eng
KW - square-free; Salié sum; asymptotic formula
UR - http://eudml.org/doc/299125
ER -

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