After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$S_{l,K_3}\left(x\right)={\sum }_{m⩽x}{M}^l\left(m\right)$$ , where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$S_{2,K_3}\left(x\right)$$ and $$S_{3,K_3}\left(x\right)$$ .

Let $K/\mathbb{Q}$ be a nonnormal cubic extension which is given by an irreducible polynomial $g\left(x\right)=x^3+a{x}^2+bx+c$. Denote by ${\zeta }_K\left(s\right)$ the Dedekind zeta-function of the field $K$ and $a_K\left(n\right)$ the number of integral ideals in $K$ with norm $n$. In this note, by the higher integral mean values and subconvexity bound of automorphic $L$-functions, the second and third moment of $a_K\left(n\right)$ is considered, i.e., $$\sum _{n\le x}{a}_K^2\left(n\right)=x{P}_1(logx)+O\left(x^{5/7+ϵ}\right),\sum _{n\le x}{a}_K^3\left(n\right)=x{P}_4(logx)+O\left(X^{321/356+ϵ}\right),$$ where $P_1\left(t\right)$, $P_4\left(t\right)$ are polynomials of degree 1, 4, respectively, $ϵ>0$ is an arbitrarily small number.

Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called pretentious view of analytic number theory. Here we study multiplicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/ℚ, we construct a natural class ${}_K$ of completely multiplicative functions whose values are dictated by Artin symbols, and we show that...