Generalized Riemann hypothesis

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 $1/2$.

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 $1/2$.

In terms of a particular L-function

All the zeros of the Dirichlet L-function for the Legendre symbol for any prime $p$ have real part $1/2$.

In terms of the prime-counting function

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

$\pi (x;n,a)={\frac {\operatorname {li} (x)}{\varphi (n)}}+O(x^{1/2+\epsilon })$.

Related facts and conjectures

Weaker conjectures

• Riemann hypothesis

Other variations of the Riemann hypothesis

• Generalization of Riemann hypothesis for number fields

External links

• A paper discussing the Riemann hypothesis and applications to computation