Sums of the triple divisor function over values of a quaternary quadratic form
Volume 183 / 2018
Acta Arithmetica 183 (2018), 63-85
MSC: Primary 11P05; Secondary 11P32.
DOI: 10.4064/aa170120-20-10
Published online: 2 March 2018
Abstract
Let $\tau_3(n)$ be the triple divisor function, the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3,n_4\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2+n_4^2)=c_1x^2(\log x)^2+ c_2x^2\log x +c_3x^2+O_{\varepsilon}(x^{7/4+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.
