Dirichlet Series - Formal Dirichlet Series

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.