ИСТИНА

The report will be devoted to two topics in which statements from number theory find an exact analogy in functional fields. Elementary number theory is based on arithmetic properties in the ring of integers Z. It turns out that many arithmetic properties are preserved if we consider ring of polynomials over the finite field Fq[x]. In the first part of the report we will discuss analogues of such classical statements, like Fermat’s, Euler’s, Wilson’s theorems, mean value theorems for traditional multiplicative functions. To develop this analogy, let us also consider the quadratic reciprocity law and Dirichlet’s theorem on prime numbers in arithmetic progressions. The second part of the report will be devoted to functional continued fractions and statements related to them, which are almost literal analogues of statements from the theory of continued fractions. In particular, the connection with analogues of Pell’s equation, analogues of Serret’s theorems and Lagrange’s theorem on the periodicity of quadratic irrationalities will be formulated. We will also discuss the recent result of U. Zannier on the periodicity of powers of incomplete quotients functional continued fractions.