In this paper the theorem on inequality connected with weighted function in method sieve weights is obtained.

An asymptotic formula is proved for the number of perfect squares in the sequence [αn] for algebraic numbers α and irrational numbers α with restricted partial quotients.

In the present article new lower and upper bounds for the convergence exponent of the special integral of Tarry’s problem are proven. The question on convergence is solved for a wide class of polynomials.

In this paper we consider hypergeometric functions and their derivatives (including with respect to parameter). By means of a new construction of homogeneous simultaneous approximations low estimate of the modulus of linear form in the values of such functions is obtained.

The estimates of the mean value trigonometrical Weyl sum in a rational number field are given in the paper [1] of Arkhipov G.I. The estimates of the mean value trigonometrical sum in real algebraic numbers is found in the present article.

In this paper we prove several theorems. Theorem 1, to assess the character sums over the continuous interval based on the use of the formula A. Postnikov and Theorem 2, for the right choice of parameters, estimates of this kind.

In this paper we obtain a new estimate of the irrationality measure of number τ = ln 7/4 .

The paper celebrates the contribution of professor Archipov to the exposition of the theory of integration on surfaces.

In this paper we generalize some B.F. Skubenko and autor results on the asymptotic distribution of integer indefinite binary quadratic forms obtained with the discrete ergodic method.

New bounds for analogs of incomplete Kloosterman sums are given.

The theorems on the distribution of absolute values of the trigonometric sum with lacunary sequence of natural numbers on short intervals are proved.

The paper studies some statistical properties of polyadic expansions of certain numbers.

For irrationalities of bounded combinatorial type it is proved that the time of ε-approximation of exact boundary of the remainder term in Hecke-Kesten problem is inversely to ε.

In the paper an explicit formula for the error term in the average mean square formula for the periodic zeta-function with rational parameter in the critical strip is obtained.