On a general divisor problem related to a certain Dedekind zeta-function over a specific sequence of positive integers

We investigate the average behavior of coefficients of the Dirichlet series of positive integral power of the Dedekind zeta-function ₊䃓 (s) of a non-normal cubic extension K₃ of Q over a certain sequence of positive integers. More precisely, we prove an asymptotic formula with an error term for the sum\ ₀_{₁^{2+a₂^2+a₃^2+a₄^2+a₅^2+a₆^2 x} (a₁, ₀_₂, a₃, a₄, a₅, a₆) ^{6}}a₊, ₊䃓 (a₁^2+a₂^2+a₃^2+a₄^2+a₅^2+a₆^2), (₊䃓 (s) ) ^k: =₍=₁^a₊, ₊䃓 (n) n^{s}.