Distribution of lattice points over the four-dimensional sphere
Let \(r_k(n)\) denote the number of representations of \(n\) by a sum of \(k\) squares of integers. If \(0< A< 1\) is a constant, then it is proved that \[ \sum_{-A\sqrt{n}\leq w\leq A\sqrt{n}} r_3(n-w^2)= \mu_4(A) r_4(n)+ O(n^\alpha), \] where \(\mu_4(A)> 0\). The author improves his former estimation with \(\alpha= \frac{3}{4}+ \varepsilon\) to \(\alpha= \frac{1487} {2000}\). He combines his own method with a method of \textit{E. P. Golubeva} [Zap. Nauchn. Semin. LOMI 144, 27-37 (1985; Zbl 0568.10013)], which was used to estimate some singular series.
asymptotic results on arithmetic functions
scientific article; zbMATH DE number 1777285