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On a boundary behavior of a class of Dirichlet series with multiplicative coefficients. (Russian. English summary) Zbl 1440.11174

Summary: In this paper we consider the behavior of functions defined by Dirichlet series with multiplicative coefficients and with bounded summatory function when approaching the imaginary axis. We show that the points of the imaginary axis are also the points of continuity in a broad sense of functions defined by Dirichlet series with multiplicative coefficients which are determined by nonprincipal generalized characters. This result is particularly interesting in connection with a solution of Chudakov’ hypothesis, which states that any finite-valued numerical character, which does not vanish on all prime numbers and has bounded summatory function, is a Dirichlet character.
The proof of the main result in this paper is based on the method of reduction to power series, basic principles of which were developed by N. V. Kuznetsov in the early 1980s. This method establishes a connection between analytic properties of Dirichlet series and boundary properties of the corresponding power series (i.e. a power series with the same coefficients as the Dirichlet series). This allows to obtain new results both for the Dirichlet series and for the power series. In our case this method allowed us to prove the main result using the properties of the power series with multiplicative coefficients determined by the nonprincipal generalized characters, which also were obtained in this work.

MSC:

11M41 Other Dirichlet series and zeta functions
30B30 Boundary behavior of power series in one complex variable; over-convergence
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References:

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