Dirichlet Series


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Dirichlet series

[‚dē·rē′klā ‚sir·ēz]
(mathematics)
A series whose n th term is a complex number divided by n to the z th power.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Dirichlet Series

 

(named for P. G. L. Dirichlet), series of the form

where the an are constants and s = σ + it is a complex variable. For example, the series

represents the zeta function for σ > 1. The theory of Dirichlet series originally arose under the strong influence of analytic number theory. Eventually it developed into an extensive branch in the theory of analytic functions.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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References in periodicals archive ?
The periodic zeta-function [zeta](s; a) is defined, for [sigma] > 1, by the Dirichlet series
On Discrete Value Distribution of Certain Compositions
In 2001, Navas [10] obtained analytic continuation of the Fibonacci Dirichlet series [[zeta].sub.F] (s).
Analytic continuation of the multiple Fibonacci zeta functions
Among the topics are quadratic points of classical modular curves, p-adic point counting on singular super-elliptic curves, a vanishing criterion for Dirichlet series with periodic coefficients, the Sato-Tate conjecture for a Picard curve with a complex multiplication, arithmetic twists with abelian extensions, and transcendental numbers with special values of Dirichlet series.
Number Theory Related to Modular Curves: Momose Memorial Volume: Seminar in Memory of Fumiyuki Momose Barcelona-Boston-Tokyo Number Theory Barcelona
where [B.sub.0](x) is given and [a.sub.0] = [[integral].sup.1.sub.0] [B.sub.0](x)dx; they used these generalized Bernoulli polynomials to derive formulas of certain Dirichlet series.
Generalized Fractional-Order Bernoulli Functions via Riemann-Liouville Operator and Their Applications in the Evaluation of Dirichlet Series
In general the theory of polynomials and multilinear operators between normed spaces has its importance in different areas of Mathematics, from Number Theory, or Dirichlet series, to Functional Analysis.
Holder's inequality: some recent and unexpected applications
The kernel function can be expressed in Dirichlet Series form [17]:
Elastic-Plastic Endochronic Constitutive Model of 0Crl7Ni4Cu4Nb Stainless Steels
Gunnells, Metaplectic ice, in Multiple Dirichlet Series, L-Functions, and Automorphic Forms, D.
Deformations of Weyl's denominator formula: six conjectures and one result
where the Dirichlet series G(s); = [[summation].sup.[infinity].sub.n = 1] g(n)/[n.sup.s][infinity] is absolutely convergent for Rs = [sigma] > 1/6.
A short interval result for the extension of the exponential divisor function
Widder (1898-1990) drew material from a course of lectures that he gave now and then at Harvard on Dirichlet series and Laplace integrals, to explain the Laplace transform for students who have a fundamental knowledge of analysis, though proofs are included for those with only a little such knowledge.
The Laplace transform. (reprint, 1941)
In this note, we will show that the set of functions [[xi].sub.[alpha]] ([alpha] [member of] C) forms an Abelian group with the Dirichlet series multiplication followed by a number of applications.
Generalized zeta functions
However, the formula may be used to construct twin prime Dirichlet series. Such a method is applied here to twin primes p,p' = p + 2D for odd D > 0 first and then even D > 0.
Generalized twin prime formulas

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