Sums of the triple divisor function over values of a ternary quadratic form

Abstract

Let $\ddot{I}{„}_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1{d}_2{d}_3=n$ in natural numbers. It is shown that $$\sum _{1\le n_1,n_2,n_3\le \sqrt{x}}\ddot{I}{„}_3(n_1^2+n_2^2+n_3^2)=c_1{x}^{\frac{3}{2}}(\log x{)}^2+c_2{x}^{\frac{3}{2}}\log x+c_3{x}^{\frac{3}{2}}+O_{\epsilon }(x^{\frac{11}{8}+\epsilon })$$ for some constants $c_1$, $c_2$ and $c_3$.

30 pages