Many mathematical and practical problems in one or another manner are related to some kind of approximation. Investigations of complicated mathematical objects or practical processes by using approximation are reduced to simpler ones. Our project is devoted to the appoximation of wide classes of analytic functions by shifts of zeta-functions depending on parameters of certain arithmetical nature, and to allied problems. In order to achieve this, new analytical and algebraic methods will be created and applied. Therefore, the main idea of the project is  the connection between algebraic and analytic approaches for solving complicated  open problems in approximation theory and some related fields. To this end, we  plan to approximate complicated analytic functions by shifts of zeta-functions with algebraic irrational parameters (Hurwitz, Lerch zeta-functions). We give  some results for the effectivization of approximation theorems by shifts of zeta-functions of holomorphic cusp forms and other zeta-functions. We use our theory in order to solve the open problems regarding the value-distribution of zeta-functions. We create new methods for the investigation of properties of algebraic numbers and their minimal polynomials and apply these properties in approximation theory and in other theoretical and practical problems (combinatorics, uniforms distribution theory, ergodic theory). We use the zero-distribution of the Riemann zeta-function and properties of Gram points in the approximation theory.
The main expected results include the approximation of a wide class of analytic functions by shifts of zeta-functions with algebraic irrational parameters (Hurwitz, Lerch, periodic Hurwitz zeta-functions), auxiliary results for the effectivization of approximation of analytic functions by shifts of zeta-functions of cusp forms, some results on the open general effectivization problem by zeta-functions. We also expect new results the distribution of roots of polynomials with integer coefficients.