# Generalized Riemann hypothesis

From Number

## Contents

## Statement

The **generalized Riemann hypothesis** can be stated in the following equivalent forms.

### In terms of L-functions

All the zeros of any Dirichlet L-function (i.e., the function obtained as the analytic continuation of the Dirichlet series of a Dirichlet character) have real part .

Note that for the Riemann zeta-function, which is not a Dirichlet L-function, the statement (called the Riemann hypothesis) is only that all the *nontrivial* zeros have real part .

### In terms of a particular L-function

All the zeros of the Dirichlet L-function for the Legendre symbol for any prime have real part .

### In terms of the prime-counting function

We have the following bound for the modular prime-counting function:

.

## Related facts and conjectures

### Weaker conjectures

### Other variations of the Riemann hypothesis