The four-dimensional divisor problem

Let $ \vec{a}_4 = (a_1,a_2,a_3,a_4)$, where $a_i$ are natural numbers with $1 \le a_1 \le a_2 \le a_3 \le a_4$. The divisor function $d( \vec{a}_4;n)$ counts the numbers of ways of expressing $n$ as the product $n = n_1^{a_1}n_2^{a_2}n_3^{a_3}n_4^{a_4}$. A new proof for the representation of the remainder term in the asymptotic formula for the summatory function of the four-dimensional divisor function is given.