Formal Dirichlet Series
A formal Dirichlet series over a ring R is associated to a function a from the positive integers to R
with addition and multiplication defined by
where
is the pointwise sum and
is the Dirichlet convolution of a and b.
The formal Dirichlet series form a ring Ω, indeed an R-algebra, with the zero function as additive zero element and the function δ defined by δ(1)=1, δ(n)=0 for n>1 as multiplicative identity. An element of this ring is invertible if a(1) is invertible in R. If R is commutative, so is Ω; if R is an integral domain, so is Ω. The non-zero multiplicative functions form a subgroup of the group of units of Ω.
The ring of formal Dirichlet series over C is isomorphic to a ring of formal power series in countably many variables.
Read more about this topic: Dirichlet Series
Famous quotes containing the words formal and/or series:
“I will not let him stir
Till I have used the approvèd means I have,
With wholesome syrups, drugs, and holy prayers,
To make of him a formal man again.”
—William Shakespeare (15641616)
“Life ... is not simply a series of exciting new ventures. The future is not always a whole new ball game. There tends to be unfinished business. One trails all sorts of things around with one, things that simply wont be got rid of.”
—Anita Brookner (b. 1928)