论文标题
关于多项式x^2+y^2+z^2+k的Re-Free值
On the r-free values of the polynomial x^2+y^2+z^2+k
论文作者
论文摘要
令k为固定整数。我们研究了R（H，R，K）的渐近公式，该公式是阳性整数溶液的数量X，Y，Z大于或等于1且小于或小于或等于H，因此多项式X^2+Y^2+Z^2+K是无R的。我们在所有大于或等于2的R（H，R，K）的渐近公式中获得了渐近公式。即使在r = 2的情况下，我们的结果也是新的。我们证明R（H，2，K）= CKH^3 + O（H^（H^（9/4 + epsilon）），其中CK> 0在k>上是不断的。这在Zhou和ding获得的误差项O（H^（7/3+Epsilon））上有所改善。
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.
