부산대학교 도서관

In 2012, D.~I. Tolev [Monatsh. Math. {\bf 165} (2012), no.~3-4, 557--567;MR2891268] studied the square-free values of a two-variable polynomial. For any $\varepsilon>0$, he proved that$$\Gamma(H)=\prod_{p}\left(1-\frac{\lambda(p^2)}{p^4}\right)H^2+O(H^{4/3+\varepsilon}),$$where $\Gamma(H)$ is the number of the square-free values of$x^2+y^2+1$ with $1\leq x,y\leq H$ and $\lambda(q)$ is the number ofthe integer solutions to the following congruence equation:$$1\leq x,y\leq q, \quad x^2+y^2+1\equiv 0\pmod{q}.$$\par Using Tolev's method and an estimate for the Salié sum, G.-L. Zhou andY. Ding [J. Number Theory {\bf 236} (2022), 308--322; MR4395352] studied the asymptotics of the number of square-free valuesrepresented by the polynomial $x^2+y^2+z^2+k$. They obtained theasymptotic formula$$\sum_{1\leq x, y,z\leqH}\mu^2(x^2+y^2+z^2+k)=c_{1}H^3+O(H^{7/3+\varepsilon}),$$where $c_{1}$ is an absolute constant.\par In the paper under review, the authors study the asymptotic formula of $r$-freevalues represented by the polynomial $x^2+y^2+z^2+k$. They alsoprovide an improvement on the error term obtained by Zhou and Ding [op.cit.].