The arithmetic sum and Gaussian multiplication theorem

The paper presents the fundamentals of the theory of arithmetic sums  and oscillatory integrals of polynomials Bernoulli, an argument that is the real function of a certain differential properties.
Drawing an analogy with the method of trigonometric sums I. M. Vinogradov.
The introduction listed problems in number theory and mathematical analysis, which deal the study of the above mentioned sums and integrals.
Research arithmetic sums essentially uses a functional equation type Gauss theorem for multiplication of the Euler gamma function.
Estimations of the individual arithmetic the amounts found indicators of convergence of their averages. In particular, the problems are solved analogues Hua Loo-Keng for one-dimensional integrals and sums.
Bibliography: 21 titles.