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

Let τ₃(n) be the triple divisor function. It is proved that ∑1≤ n₁,n₂,n₃≤ √xτ₃(n₁²+n₂²+n₃²)=c₁x3/2(log x)²+ c₂x3/2log x +c₃x3/2 +Oε(x13/10+ε) for some constants c₁, c₂ and c₃, updating a result of the second author and Zhang. Moreover, we show that ∑1≤ n₁,n₂,n₃,n₄≤ √xτ₃(n₁²+n₂²+n₃²+n₄²) =c₄x²(log x)²+c₅x²log x+c₆x² +Oε(x5/3+ε) for some constants c₄, c₅ and c₆, which improves the results in [6].