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