ON THE BOUNDARY BEHAVIOR OF A CLASS OF DIRICHLET SERIES

Abstract

In this paper we study the problem of analytical behavior of Dirichlet series with a bounded summatory function on its axis of convergence, σ = 0. This problem was also considered in the authors’ earlier works in case of Dirichlet series with coefficients determined by finite-valued numerical characters, which, in turn, was connected with a solution for a well-known Chudakov hypothesis.

The Chudakov hypothesis suggests that generalized characters, which do not vanish on almost all prime numbers p and asymptotic behavior of whose summatory functions is linear, are Dirichlet characters. This hypothesis was proposed in 1950 and was not completely proven until now. A partial proof based on the behavior of a corresponding Dirichlet series when it approaches to the imaginary axis was obtained in one of authors’ works. There are reasons to anticipate that this approach may eventually lead to a full proof of the Chudakov hypothesis.

In our case this problem is particularly interesting in connection with finding analytical conditions of almost periodic behavior of a bounded number sequence, different from those obtained before by various authors, for example, by Szego.

Our study is based on a so called method of reduction to power series. This method was developed by Prof. V. N. Kuznetsov in the 1980s and it consists in studying the relation between the analytical properties of Dirichlet series and the boundary behavior of the corresponding (i.e. with the same coefficients) power series.

In our case this method of reduction to power series allowed us to show that such Dirichlet series are continuous in the wide sense on the entire imaginary axis. Moreover, this method also helped to construct a sequence of Dirichlet polynomials which converge to a function determined by a Dirichlet series in any rectangle inside the critical strip.