Let $\Lambda_k(n)=\sum_{d|n}\mu(d)\log^k\frac{n}{d}$, where $\mu(d)$ is the Möbius function. Then the following is proved: a) $\Lambda_2(n)\log =\Lambda_1(n)\log^2n+2\sum_{d|n}\Lambda_1(d)\Lambda_1\big(\frac nd\big)\log d$, b) $\Lambda_2(n)\leq2\Lambda_1(n)\log n+\frac12\log^2n-\Lambda^2_1(n)$. c) For every $m$ less than $k$ \[ ambda_k(n)=ambda_{k-m}(n)og^mn+um_{i=0}^{m-1}\binom{m}{i}um_{d|n}ambda_{k-m}(d)ambda_{m-i}\Big(\frac nd\Big)og^id. \] d) $\Lambda_1(n)\log^{k-1}n\leq\Lambda_kn\leq k\log^kn$. e) $-\rho(x)\log^kx=\sum_{n\leq x}\Lambda_1(n)\log^{k-1}n\rho\big(\frac xn\big)+O(x\log^{k-1}x),\quad k\geq2$, oindent where \[ \rho(x)=si_1(x)-x \] and \[ si_1(x)=um_{neq x}ambda_1(n). \]