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

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.