Editing Generalized Riemann hypothesis (section)

{{Short description|Mathematical conjecture about zeros of L-functions}}
The [[Riemann hypothesis]] is one of the most important [[conjecture]]s in [[mathematics]]. It is a statement about the zeros of the [[Riemann zeta function]]. Various geometrical and arithmetical objects can be described by so-called global [[L-function|''L''-function]]s, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the [[algebraic function field]] case (not the number field case).

Global ''L''-functions can be associated to [[elliptic curve]]s, [[number field]]s (in which case they are called [[Dedekind zeta-function]]s), [[Maass form]]s, and [[Dirichlet character]]s (in which case they are called [[Dirichlet L-series|Dirichlet L-function]]s). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the '''extended Riemann hypothesis (ERH)''' and when it is formulated for Dirichlet ''L''-functions, it is known as the '''generalized Riemann hypothesis''' or '''generalised Riemann hypothesis''' (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label ''generalized Riemann hypothesis'' to cover the extension of the Riemann hypothesis to all global ''L''-functions, 
not just the special case of Dirichlet ''L''-functions.)