# Mathematics

## On an inverse problem for differential operators on hedgehog-type graphs

An inverse spectral problem is studied for Sturm–Liouville differential operators on hedgehog-type graphs with generalized matching conditions in the interior vertices and with Dirichlet boundary conditions in the boundary vertices. A uniqueness theorem of recovering potentials from given spectral characteristics is provided, and a constructive solution for the inverse problem is obtained.

## About harmonic analysis of periodic at infinity functions

We consider slowly varying and periodic at infinity multivariable functions in Banach space. We introduce the notion of Fourier series of periodic at infinity function, study the properties of Fourier series and their convergence. Basic results are derived with the use of isometric representations theory.

## On a Particular Equivalent of Extended Riemann Hypothesis for Dirichlet L-functions on Numerical Fields

A condition on summatory function over a set of prime ideals for Dirichlet L-functions on numerical fields is obtained. This condition is equivalent to extended Riemann hypothesis. Analytical properties of Euler products associated with this equivalent are studied

## To the Problem of the Integrity of the Artin’s L-functions

In this paper was described a class of Artin’s L-functions, each of which is meromorphic, their poles lays on the critical line Re s = 1/2 and coincides with zeroes of Dedekind’s Z-functions of some fields.

## Inverse problem for Sturm–Liouville operator on the half-line having nonintegrable singularity in an interior point

The inverse problem of recovering Sturm–Liouville operators on the half-line with a nonintegrable Bessel-type singularity in an interior point from the given Weyl function is studied. The corresponding uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided. Necessary and sufficient conditions of the solvability of the inverse problem are obtained.

## Necessary and sufficient conditions of belonging to the Besov–Potapov classes and Fourier coefficients with respect to multiplicative systems

In this paper we obtain necessary and sufficient conditions for a function to belong to the Besov–Potapov classes. Using functions with Fourier coefficients with respect to multiplicative systems from the class GM, we show the sharpness of some these results.

## Mathematical life of G. I. Arkhipov

This paper presents the most important discoveries made by outstanding mathematician G. I. Arhkipov since the end of 60s to the middle of 2000s.

## Полупростые градуированные кольца

The graded version ofWedderburn–Artin theorem is obtained. It gives description of semisimpleG-graded ring for arbitrary groupG. Homological classification of graded semisimple rings is given.

## An Estimate of a Certain Summatory Functions Class

In this paper summatory functions of form P n·x h(n)nit, 1 · |t| · T for finite-valued functions h(n) of natural argument with bounded sum function are estimated

## Approximation Polynomials and Dirichlet L-functions Behavior in the Critical Strip

In this paper a sequence of Dirichlet polynomials that approximate Dirichlet L-functions is constructed. This allows to calculate zeros of L-functions in an effective way and make an assumptions about Dirichlet L-function behavior in the critical strip.