Boundary behavior and the problem of analytic continuation of a certain class of Dirichlet series with multiplicative coefficients as an integral functions on the complex plane

V. N. Kuznetsov ; O. A. Matveeva

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 124-137

Résumé

The paper considers the class of Dirichlet series with multiplicative coefficients defining Functions regular in the right half-plane of the complex plane and admitting Approximation by Dirichlet polynomials in the critical strip. It is shown that the regularity condition on the imaginary axis allows one to analytically continue such series as entire functions on the complex plane. The proof of this fact is based on the properties of approximation Dirichlet polynomials and the Riemann-Schwartz ideas, embedded in the symmetry principle of analytic continuation functions of a complex variable. The class of Dirichlet series for which Analyticity analysis on the imaginary axis. It should be noted that the result obtained in the work has a direct relation to the solution of the well-known problem of generalized characters posed by Y. V. Linnik and N. G. Chudakov in the 1950s. The approach indicated in the paper in the problem of analytic continuation of Dirichlet series with numerical properties admits a generalization to Dirichlet series with characters of numeric fields. This encourages credit continuation without using the functional equation of the Dirichlet $L$-functions of numeric fields on the complex plane. We also note that the class of Dirichlet series studied in this paper belongs to the Dirichlet series whose coefficients are determined by non-principal generalized characters. It can be shown that for these series the condition of analytic continuation. As far back as 1984, V. N. Kuznetsov showed that in the case of an analytic continuation of such series in an integral way onto the complex plane determined by the order of growth of the module, then Chudakov's hypothesis that the generalized character is a Dirichlet character will take place. But the final solution of the problem of generalized characters, put in 1950 by Y. V. Linnik and N. G. Chudakov, will be given in the following papers of the authors.

Détail

Keywords: approximation Dirichlet polynomials, the Riemann-Schwarz symmetry principle, conformal mappings.