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 Chebyshevskii Sb., 2016, Volume 17, Issue 3, Pages 125–134 (Mi cheb501)

On a boundary behavior of a Dirichlet series class with multiplicative coefficients

V. N. Kuznetsov, O. A. Matveeva

Saratov State University

Abstract: In this paper we consider the behavior of funcions 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 its connection with a solution of Chudakov hyphotesis, 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 prof. Kuznetsov in the early 1980s. Ths method establishes a connection between analytical 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.
Bibliography: 16 titles.

Keywords: Dirichlet series, summatory function of the coefficients, generalized characters, Dirichlet characters.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00399_à

Full text: PDF file (566 kB)
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UDC: 511.3
Accepted:13.09.2016

Citation: V. N. Kuznetsov, O. A. Matveeva, “On a boundary behavior of a Dirichlet series class with multiplicative coefficients”, Chebyshevskii Sb., 17:3 (2016), 125–134

Citation in format AMSBIB
\Bibitem{KuzMat16} \by V.~N.~Kuznetsov, O.~A.~Matveeva \paper On a boundary behavior of a Dirichlet series class with multiplicative coefficients \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 3 \pages 125--134 \mathnet{http://mi.mathnet.ru/cheb501} \elib{https://elibrary.ru/item.asp?id=27452086}