Sergeev S. Trigonometric sums method in the asymptotic problems of number theory

Thesis is devoted to the estimation of Kloosterman sums analogues, weighted by special kind of multiplicative functions over the ring of Gaussiam integers. Kloosterman sums over norm subgroup were considered for the first time. Non-trivial estimates were obtained. Analogues of the Smarandache function over the ring of Gaussian integers were studied and estimated. Kloosterman sums were estimated over norm group for the first time. Asymptotic estimations of weighted Kloosterman sums over the rings of rational integers and Gaussian integers was also obtained. We also got estimations in narrow sectors. The asymptotic estimations were obtained for the mean value of analogue of the Smarandache function and its dual over the ring of Gaussian integers, estimates of non-symmetric Kloosterman sums, 2-dimensional Kloosterman sums, twisted Kloosterman syms and character sums over the norm groups.