Sums of the triple divisor function over values of some quadratic forms

Chenhao Du, Qingfeng Sun

Published 2024-04-21Version 1

Let $\tau_3(n)$ be the triple divisor function. It is proved that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{13}{10}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$, updating a result of the second author and Zhang. Moreover, we show that $$ \sum_{1\leq n_1,n_2,n_3,n_4\leq \sqrt{x}}\tau_3(n_1^2+n_2^2+n_3^2+n_4^2) =c_4x^{2}(\log x)^2+c_5x^{2}\log x+c_6x^{2} +O_{\varepsilon}\left(x^{\frac{5}{3}+\varepsilon}\right) $$ for some constants $c_4$, $c_5$ and $c_6$, which improves the results in [6].