# Riemann zeta-function

## Contents

## Definition

### As a Dirichlet series

The **Riemann zeta-function** is the following Dirichlet series of :

.

In other words, it is the Dirichlet series for the all ones function .

It can also be expressed as a product, using the product formula for Dirichlet series of completely multiplicative function:

.

### As the function obtained by analytic continuation

The Dirichlet series for the Riemann zeta-function is absolutely convergent for all complex numbers for which . Although the series does not make sense for other , the function extends to a meromorphic function of , with a single simple pole at the point .

### In terms of the Dirichlet eta-function

The Riemann zeta-function can be defined in terms of the Dirichlet eta-function:

.

The Dirichlet eta-function is also defined in terms of a Dirichlet series, but this Dirichlet series has the advantage of being convergent on a wider range.

## Related functions

### Similarly defined functions

- Dirichlet L-function
- Dedekind zeta-function is a generalization to number fields.

## Zeros and poles

### Poles

The Riemann zeta-function has a single simple pole at .

### Zeros

The Riemann zeta-function has zeros at all negative even integers. There are also infinitely many zeros with real part . The Riemann hypothesis states that all zeros other than the negative even integers have real part .