The periodic zeta-function [zeta](s; a) is defined, for [sigma] > 1, by the Dirichlet series
In 2001, Navas  obtained analytic continuation of the Fibonacci Dirichlet series
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
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
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.
The kernel function can be expressed in Dirichlet Series
Gunnells, Metaplectic ice, in Multiple Dirichlet Series
, L-Functions, and Automorphic Forms, D.
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.
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.
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.
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.