Lattice Points in the Circle and Sphere

Let P(x) and P3(x) be the error terms in the Gaussian circle problem and the sphere problem, respectively.We investigate the asymptotic behavior of the sums ∑k≤0k≡0modpPk,∑k≤xk≡0modpP3k.$$ {\displaystyle \sum_{\underset{k\equiv 0\left( \mod p\right)}{k\le 0}} P(k),}\;{\displaystyle \sum_{\underset{k\equiv 0\left( \mod p\right)}{k\le x}}{P}_3(k).} $$ Here, p ≥ 2 is a prime number. Bibliography: 20 titles.