On the divisor problem for values of a ternary cubic form (Q1584121)

This paper is devoted to the derivation of an asymptotic formula for the mean value \(V_k(x)\) of the function \(\tau_k(n)\) under the conditions \(1\leq n\leq x\), where \(n\) runs over the values of the following ternary cubic form \[ \varphi= \varphi(z_1,z_2, z_3)=z_1^3 +z^3_2+z^3_3-3z_1 z_2 z_3 \] where \(z_1,z_2,z_3\) are integers. Note that the value \(V_k(x)\) equals the number of solutions of the following Diophantine equations \[ x_1\dots x_k-z_1^3-z_2^3-z^3_3+3z_1z_2z_3=0 \] where the variables \(x_1,\dots,x_k\) take natural values, the variables \(z_1,z_2, z_3\) take integer values, and the inequality \(x_1\dots x_k\leq x\) holds.