On the problem of generalized characters

V. N. Kuznetsov ; O. A. Matveeva

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 210-218

Résumé

The problem of generalized characters lies in the solution of Linnik's problem, posed by him in 1949, with respect to the analytic continuation of entire functions to the complex plane of a class of Dirichlet series and in the solution of the hypothesis of N. G. Chudakov, who put forward in 1950 that any finite-valued numerical character, different from zero on almost all prime numbers and having a bounded summation function, is a Dirichlet character. Later such characters were called non-principal generalized characters. The coefficients of the Dirichlet series in Linnik's problem were also determined by non-principal generalized characters. Except Yu. V. Linnik and N. G. Chudakov's solutions to the problem of generalized characters were handled by such well-known mathematicians as V. G. Sprindzhuk, K. A. Rodossky, B. M. Bredikhin and many others, but the problem remained open. In recent years, the authors have developed an approximation approach based on the approximation in the right half-plane of the complex plane of functions given by Dirichlet series by Dirichlet polynomials in the problem of analytic continuation of Dirichlet series with multiplicative coefficients. Earlier this approach allowed the authors to solve the problem of Yu. V. Linnik, and in this paper the solution of the hypothesis of N. G. Chudakov is given.

Détail

Keywords: generalized character, approximate Dirichlet polynomials, the problem of generalized characters.