# Analytic Number Theory/Dirichlet series

For the remainder of this book, we shall use *Riemann's convention of denoting complex numbers*:

## Definition[edit | edit source]

**Definition 5.1**:

Let be an arithmetic function. Then the **Dirichlet series associated to ** is the series

- ,

where ranges over the complex numbers.

## Convergence considerations[edit | edit source]

**Theorem 5.2 (abscissa of absolute convergence)**:

Let be an arithmetic function such that the series of absolute values associated to the Dirichlet series associated to

neither diverges at all nor converges for all . Then there exists , called the **abscissa of absolute convergence**, such that the Dirichlet series associated to converges absolutely for all , and it's associated series of absolute values diverges for all , .

**Proof**:

Denote by the set of all real numbers such that

diverges. Due to the assumption, this set is neither empty nor equal to . Further, if , then for all and all , since

and due to the comparison test. It follows that has a supremum. Let be that supremum. By definition, for we have convergence, and if we had convergence for we would have found a lower upper bound due to the above argument, contradicting the definition of .

**Theorem 5.3 (abscissa of conditional convergence)**:

## Formulas[edit | edit source]

**Theorem 8.4 (Euler product)**:

Let be a strongly multiplicative function, and let such that the corresponding Dirichlet series converges absolutely. Then for that series we have the formula

- .

**Proof**:

This follows directly from theorem 2.11 and the fact that strongly multiplicative strongly multiplicative.