In this paper we consider an extension to friable integers of the arcsine law for the mean distribution of the divisors of integers, originally due to Deshouillers, Dress and Tenenbaum.We describe the limit law and show that it departs from the arcsine law when the friability parameter $u:=logx/logy$ increases. More precisely, as $u\to \infty$, the mean distribution shifts from the arcsine law towards Gaussian behaviour.

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.