The number of primitive unassociated third-order matrices of a given determinant

Abstract

When studying questions of the asymptotic distribution of integer points over domains on hyperboloids, as well as integer matrices of the second and third orders, it becomes necessary to use primitive unassociated matrices of the second and third orders of a given determinant. Counting the number of integer matrices of the same order and a given determinant requires the selection of pairwise unassociated matrices among them. Non-associated second-order matrices appear when considering preliminary ergodic theorems for flows of integer points on hyperboloids when applying the discrete ergodic method to the problem of representing integers by ternary quadratic forms. The number of unassociated second-order matrices is also used to express the number of binary quadratic forms, the arithmetic minimum of which is divisible. In addition, formulas for the number of primitive unassociated matrices of the second and third orders make it possible to determine the orders of the principal terms in asymptotic formulas for the number of integer matrices of large norm(determinant). In this paper, based on the canonical triangular form of the third-order integer matrices, a formula is obtained for the number of primitive unassociated third-order matrices represented by the canonical decomposition. A formula is also obtained for the number of primitive unassociated matrices of the third order of a given determinant, divisible by a given matrix. The main results related to the question of the number of non-associated integer matrices of a given determinant belong to Yu. V. Linnik, B. F. Skubenko, A.V. Malyshev and the authors of this work, the results of which can be further transferred to integer matrices of any order.