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

TL;DR

This paper derives an asymptotic formula for counting solutions to x^2 + y^2 + z^2 + k that are r-free, improving error bounds and covering all r ≥ 2, including the case r=2.

Contribution

It provides a new asymptotic formula for R(H, r, k) for all r ≥ 2, with improved error terms, even in the case r=2.

Findings

Asymptotic formula for R(H, r, k) for all r ≥ 2

Improved error term O(H^(9/4+ε)) for r=2

Results are new even for r=2 case

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 x, y, z greater than or equal to 1 and less than or equal to 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 greater than or equal to 2. Our result is new even in the case r = 2. We proved that R(H, 2, k) = ckH^3 + O(H^(9/4+epsilon)), where ck > 0 is a constant depending on k. This improves upon the error term O(H^(7/3+epsilon)) obtained by Zhou and Ding.