On the problem of analytical continuation of Dirichlet series with finite coefficients as entire functions onto the complex plane

O. A. Matveeva ; V. N. Kuznetsov

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 286-296

Résumé

One well-known approach to the problem of analytic continuation of Dirichlet series is analysis of properties of a sequence of primitive integrals, which arise in iterations of a summatory function of the coefficients of these series. With this approach it was possible to obtain an analytic continuation of the Riemann zeta function and Dirichlet $L$-functions. In 1975 N. G. Chudakov presented necessary and sufficient conditions for an analytic continuation of Dirichlet series as meromorphic functions with a finite Lindelöf function, expressed through behavior of primitive integrals. In this paper we formulate necessary and sufficient conditions of analytic continuation of Dirichlet series with finite-valued coefficients to an entire function. These conditions are expressed in terms of behavior of Cesàro means of coefficients of a Dirichlet series. Unlike the result of N. G. Chudakov, where conditions of analytic continuation are expressed as an existence theorem, in this paper we obtain an explicit form of the asymptotics of Cesàro means. This result is based on the approximation approach developed earlier by V. N. Kuznetsov and the author, which made it possible to establish a connection between the solution of this problem and a possibility to approximate entire functions defined by Dirichlet series by Dirichlet polynomials in the critical strip.

Détail

Keywords: Dirichlet series, analytic continuation, joint approximation of a function and its derivatives.