On The Square-free Values Of A Class Of Ternary Quadratic Polynomials

This paper involves the asymptotic estimate about square-free values and other relevant problems.In 2012,Tolev considered the polynomial x2+y2+1 and established the asymptotic formula of the number of pairs of integers 1≤x,y≤H such that it is square-free.Let S3(H,k)be the number of integer solutions satisfying 1≤x,y,z≤H such that the polynomial x2+y2+z2+k(k is a fixed integer)is square-free.In 2022,Zhou and Ding obtained the asymptotic formula of S3(H,k)with the error term O(H7/3+ε)by using Tolev’s method and Salié sum.In this paper,we improve Zhou and Ding’s result,and prove that S3(H,k)=ckH3+O(H9/4+ε),where ck is a constant depending on k.In addition,we generalize the study about square-free values and obtain more general results.Let R(H,r,k)be the number of positive integer solutions x,y,z ≤H such that the polynomial x2+y2+z2+k is r-free.We give the asymptotic formula of R(H,r,k)for all r≥2 and all integers k.Our result is as follows.For k≠0,(?),and(?),where cr,k is a constant depending on r and k.