On the square-free values of the polynomial x2 + y2 + z2 + k

Abstract

Square-free values of polynomials had been studied by various authors, including Estermann, Heath-Brown and Hooley. For 1≤x,y≤H, Tolev proved that the number of the square-free values attained by the polynomial x2+y2+1 has the asymptotic formula c1H2+O(H4/3+ε), where is c1 is an absolute constant and ε is an arbitrary small positive number. The key ingredient of his proof which leads to the elaborate error term is the estimate for the Kloosterman sum. In this paper, by using Tolev's method and some estimate for the Salié sum, we show that for any fixed integer k, there is an absolute constant c2 such that the number of square-free values of the polynomial x2+y2+z2+k with 1≤x,y,z≤H is c2H3+O(H7/3+ε).