Analogue of the Duffin – Scheffer theorem for one class of Dirichlet series with finite-valued coefficients

Abstract

The well-known theorem, proved by Doffin and Scheffer, states that the boundedness of a power series with finite-valued coefficients in a certain sector of the unit circle is equivalent to the periodicity of its coefficients, starting from a certain number. The paper indicates the class of Dirichlet series with finite-valued coefficients bounded in any strip of the right half-plane of the complex complex plane by a constant depending only on the height of the strip, for which an analogue of Dauffin – Scheffer theorem is proved. Earlier, an analogue of the Dauffin – Scheffer theorem was obtained by the authors for Dirichlet series with multiplicative coefficients. The method of proving this result allowed, in particular, to solve the well-known problem of generalized characters posed in 1950 by Yu.V. Linnik and N.G. Eccentric In this paper, this technique is used to prove an analogue of the Duffin – Scheffer theorem for the indicated class of Dirichlet series with multiplicative coefficients.