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 Lindelo¨f 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 Cesa`ro 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 Cesa`ro 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.
Highlights
Chudakov, where conditions of analytic continuation are expressed as an existence theorem, in this paper we obtain an explicit form of the asymptotics of Cesaro means
This result is based on the approximation approach developed earlier by V
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
Summary
+ it является изучение свойств сумматорной функции коэффициентов и ее итераций, возникающих в результате последовательного интегрирования по частям интеграла в известном интегральном представлении ряда Дирихле в области сходимости:. |f (s)Γ(s)| < e−α|t|, 0 ≤ σ ≤ 1, где Γ(s) — гамма-функция, α > 0 и зависит от функции f (s), когда для любого m определяется алгебраический полином Tm−1(x) степени m − 1, такой, что для чезаровских средних порядка m от коэффициентов ряда Дирихле при любом x ≥ 1 выполняются равенства. Тогда соответствующий (с теми же коэффициентами, что и у ряда Дирихле) степенной ряд определяет функцию g(x) = ∑︁ akxk, k=1 имеющую производные любого порядка на отрезке [0, 1], и для любого m существует последовательность полиномов Pn(x), n. В [3], [4] показано, что если ряд Дирихле (2) определяет целую функцию с условием (3), то соответствующий степенной ряд (5) имеет конечные односторонние производные любого порядка (6) в точке единица, то есть степенной ряд (5) определяет функцию g(x), имеющую производные любого порядка на отрезке [0, 1]. Пусть также Snm(ν) — чезаровские средние порядка m для коэффициентов многочленов (8)
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