# Dirichlet series

From Number

## Contents

## Definition

Suppose is a function. In other words, is an arithmetic function. The **Dirichlet series** of is defined as the following *formal series*:

.

The corresponding function is defined as the sum of the series where convergence of the summation is considered in . Even though the series may converge on only a part of , it might be possible to extend to a global meromorphic function, or to a meromorphic function on a large part of .

We sometimes distinguish between the *Dirichlet series* (which is the formal series) and the *Dirichlet function* (which is the function obtained using the series and taking analytic continuation to parts of the complex plane where the series is not defined).

## Facts

### Nice forms for Dirichlet series of certain kinds of functions

- Euler product formula for Dirichlet series of multiplicative function
- Euler product formula for Dirichlet series of completely multiplicative function