Generalized Riemann hypothesis

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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


External links