Sinyavs'kyj O. The generalized divisors functions

In a ring of the Gaussian integers the asymptotic formula summatory function for the divisors function weighed by the exponential function or a by sum of the Ramanujan, or by generalized sum Kloosterman is constructed. The asymptotic formula summatory function for the function of a sum of divisors of the Gaussian integers on an arithmetic progression and in narrow sector is constructed. There are using the representation of the function of a sum of divisors as a series on sums of the Ramanujan. The asymptotic formula for summatory functions of number of representations natural n as product of the sum of two squares and number which does not contain a square simple and is compared with l on the module q is found. Using a method exponential pairs, the new estimation of an error term in the many-dimensional asymmetrical problem of divisors is obtained. The asymptotic formula for summatory function for the many-dimensional asymmetrical function of divisors on guadrates natural n is received.