Let $K_3/{\mathbb{Q}}$ be a non-normal cubic extension, which is given by anirreducible polynomial $g(x) = x^3 + ax^2 + bx + c$. In this paper, we study the meanvalue estimates related to the Dedekind zeta-function $\xi_K_3 (s)$. We first introduce a new ingredient to improve a previous result of L\"{u} (Open Math. 11(2) (2012) 274--282). Moreover, we consider a generalized divisor problem and establish asymptotic formulas.

Suppose that $\mathbb{H}^∗$ is the set of all primitive cusp forms $f$ of even integral weight $k \geq 2$ for the full modular group $SL_2(\mathbb{Z})$. In this paper, we establish asymptotic formulas for the second moment of Fourier coefficients of the triple product $L$-function $L(s, f \otimes f \otimes f )$ and the related $L$-function $L(s, {\rm sym}^2 f \otimes f)$ attached to $f$ on average, which improves previous results.