On a problem of Yu. V. Linnik

In the late 40-s of the last century Yu. V. Linnik posed the problem of analytic continuation in an integral way to the complex plane of Dirichlet series whose coefficients are determined by finite-valued numerical characteristics that are not equal to zero on almost all primes and have bounded summation functions. Such characters are called non-principal generalized characters. Many mathematicians dealt with Linnik's problem. In particular, this task was handled by N. G. Chudakov, who saw the way to solving it in proving his suggestion that the non-principal generalized character is the Dirichlet character.
Linnik's problem and the hypothesis of N. G. Chudakov is widely known in number theory. It is called the problem of generalized characters.
In this paper we solve the problem of Yu. V. Linnik, based on the results obtained earlier by the authors concerning the analytic continuation of Dirichlet series with multiplicative coefficients.
Thus, in this paper a partial solution of the generalized character problem posed in the 1950-s Yu. V. Linnik and N. G. Chudakov.