Dadayan Z. Pillai's function and its generalizations

For the first time Pillai's function is considered over the ring of Gaussian integers, for the summatory function of which the asymptotic formulas are constructed. Also the asymptotic estimates are obtained for the summatory functions, in which the generalized Pillai's function is weighted by degree norm of its argument. The asymptotic estimations are obtained for the mean values of Pillai's function and its generalization, when the arguments of these functions run the arithmetic progressions. Also asymptotic estimation for the second moment of the remainder member of the summatory function with weighted generalized Pillai's function over the ring of Gaussian integers is constructed.