Velychko I. Arithmetical functions over the matrix rings.

The thesis is devoted to еру problems of construction and research of arithmetical functions over the matrix rings with integer elements and trigonometrical sums over the matrix rings with integers and gaussian integers. The dissertation is mainly focused on the problem of searching a distribution of defined functions, most of which are related to the classical divisor function . Specifically, asymptotic formulas for appropriate summatory functions and estimates for the error terms have been obtained for analogs of the divisor function over the matrix rings of the order 3 and 4. Besides, a distribution over the set of square-free numbers has been found for the generalized divisor function . The analog of the Piltz's problem over the matrix ring of the second order has been considered, an appropriate asymptotic formula and an estimate of the second moment of the error term have been obtained. In the second part of the thesis we have introduced the congruence relation, residue system and inverse element modulo matrix over the matrix ring of any fix order. Using these notions, we have introduced the notation of the Kloosterman sums over the ring with integers and given the non-trivial estimates for such sums when the order of the ring is equal to 2 or 3. The notions of the congruence relation, residue system and inverse element modulo matrix have been extended over the matrix ring with gaussian integers. Using these notions, we have introduced the notation of the Kloosterman sums over the ring with gaussian integers and given the non-trivial estimate for such sums when the order of the ring is equal to 2.