In order to see this (the convergence of the infinite product), let’s instead observe our other main example of a sum over divisors equalling a product over primes working. When we compared them for \(\phi\) above, we got
We give such series a name. The following definition is purely formal, considered without considering issues such as convergence. (See [E.2.8, Chapter 4.6] for an interesting formal viewpoint on the set of these series.)
Definition24.3.1.Dirichlet Series.
In general, for an arithmetic function \(f(n)\text{,}\) its Dirichlet series is
Answer the following three questions to see if you understand this definition. (See Exercise 24.7.1.)
For what arithmetic function is the Riemann zeta function the Dirichlet series?
What would the Dirichlet series of \(N\) be?
What about the Dirichlet series of \(I\text{?}\)
Note that this already indicates some level of connection between arithmetic functions. These are connections which may not have been evident otherwise.
Subsection24.3.2Euler products
For our purposes, the very important thing to note about such series is that they often can be expanded as infinite products.
Definition24.3.2.Euler Products.
In general, for an arithmetic function \(f(n)\text{,}\) its Dirichlet series is said to have an Euler product if the series can be written as an infinite product in the following manner:
\begin{equation*}
\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p (\text{ a formula involving }f(p)\text{ and }p^s)\text{.}
\end{equation*}
Example24.3.3.Euler product for Riemann zeta function.
We have already suggested one for the zeta function:
In the next section, we justify more of this discussion, and connect our wonderful results about Dirichlet products of finite arithmetic functions to deep properties of their Dirichlet series.