It is obtained the unique existence theorem of solutions to the mixed problem for linear hyperbolic systems of second order with inhomogeneous Neumann boundary condition. And, it is investigated that the constants appearing in the first energy inequality depend essentially only on $\mathscr{B}^{1+\mu}$-norms $(0\lt\mu\ll 1)$ of coefficients of operators.

Let $E$ designate either of the classical error terms for the summatory functions of the arithmetical functions $\phi(n)/n$ and $\sigma(n)/n$ ($\phi$ is Euler's function and $\sigma$ the divisor function). By following an idea of Codecà's [3] and by refining some of his estimates we prove that $|E|$ has asymptotic $\lambda$-th order means for all positive real numbers $\lambda$. We also prove that $E$ has asymptotic $k$-th order means for all positive integers $k$, and that this mean is zero whenever $k$ is odd. The results obtained can be applied to functions other than $E$ as well, such as the functions $P$ and $Q$ of Hardy and Littlewood[8], or the divisor functions $G_{-1,k}$[9].