Can you give an example of an irreducible element of the ring of Dirichlet series with integer coefficients?

Question

According to this. The ring of Dirichlet series with integer coefficients is a UFD. Can you give an example of an irreducible element in that ring?

Answer 1

Let $A$ be the set of arithmetic functions, and let $U$ be the set of units so that $f\in A$ iff $f(1)\neq 0$.

Consider the map $\theta :A\rightarrow \mathbb{Z}:\theta(f)=n$, where $n$ is the smallest integer such that $f(n)\neq 0$.

For example, let $f_p(n)=1$ iff $n=p$ and $0$ everywhere else, so it is easily seen that $\theta(f_p)=p$. A function $f$ is a unit iff $f(1)\neq 0$, and therefore $f$ is a unit if and only if $\theta(f)=1$.

Now by a relatively simple calculation, it may be shown that $\theta(f\star g)=\theta(f)\theta(g)$ $\forall f,g\in A$. This means that you can use the irreducibles of $\mathbb{Z}$ (the primes) to show that elements of $A$ are irreducible.

Suppose $f_p=a\star b$, then one has that $p=\theta(f_p)=\theta(a)\theta(b)$. Therefore wlog $\theta(a)=1$ and $a\in U$, thus proving that $f_p$ is irreducible.

What's more, given arbitrary $g\in A$ with $\theta(g)=p_1^{k_1}\ldots p_l^{k_l}$ there exists a unique unit function $u\in U$ such that $g=u\star f_{p_1}^{k_1}\star\ldots\star f_{p_l}^{k_l}$. This is a unique product of irreducibles (uniqueness by a similar process to that of the integers) and shows that a function is irreducible if and only if $\theta$ maps it to a prime.