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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.

Recently, Solomon has introduced a zeta function which counts sublattices of a given lattice over an order ([5]). Let us recall the definition of this zeta function. Let I be a (finitedimensional) semisimple algebra over the rational fieldQ or over the j!?-adicfieldQp, and let A be an order in I. A is a Z-order when I is a Q-algebra, while A is a Zp-order when 2 is a Qp-algebra, where Zp is the ring of p-adic integers. Throughout this paper, p stands for a rational prime and the subscript p indicates the p-dAic completion. Let V be a finitelygenerated left I'-module, and let L be a full//-lattice in V. Solomon's zeta function is defined as