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This article is cited in 4 scientific papers (total in 4 papers)
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 Saratov State University
Abstract:
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.
Keywords:
Dirichlet series, analytic continuation, joint approximation of a function and its derivatives.
Received: 01.09.2017 Accepted: 14.12.2017
Citation:
O. A. Matveeva, V. N. Kuznetsov, “On the problem of analytical continuation of Dirichlet series with finite coefficients as entire functions onto the complex plane”, Chebyshevskii Sb., 18:4 (2017), 286–296
Linking options:
https://www.mathnet.ru/eng/cheb612 https://www.mathnet.ru/eng/cheb/v18/i4/p286
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Abstract page: | 172 | Full-text PDF : | 70 | References: | 19 |
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