# Riemann hypothesis, generalized

A statement about the non-trivial zeros of Dirichlet $L$-functions (cf. Dirichlet $L$-function), Dedekind zeta-functions (cf. Zeta-function) and several other similar functions, similar to the Riemann hypothesis (cf. Riemann hypotheses) on the non-trivial zeros of the Riemann zeta-function $\zeta ( s)$. In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Riemann hypothesis.

For Dirichlet $L$-functions it is not even known whether there exist real zeros in the interval $[ 0, 1]$ (Siegel zeros). This is important in connection with the class number of quadratic fields (see also Quadratic field; Siegel theorem).

Let $K$ be an algebraic number field, $G( K)$ the group of fractional ideals of $K$ and $C( K)$ its idèle class group (cf. Idèle; Fractional ideal). Let $X$ be a quasi-character on $C( K)$, i.e. a continuous homomorphism of $C( K)$ into the group of non-zero complex numbers. Then for an idèle $( x _ {v} )$ one has $X ( ( x _ {v} ) ) = \prod _ {v} X _ {v} ( x _ {v} )$, where for each $v$, $X _ {v}$ is a quasi-character of $K _ {v} ^ {*}$ which is equal to unity on $U( K _ {v} )$, the units of the local completion $K _ {v}$, for almost-all $v$. Let $S$ be a finite subset of the valuations on $K$ including the Archimedian ones, $S _ \infty$. A function $\chi$ can now be defined on $G( K)$ by setting for all prime ideals $\mathfrak P$,

$$\chi ( \mathfrak P ) = \left \{ \begin{array}{cl} {X _ {v} ( \mathfrak p v) } & {\textrm{ if } \mathfrak P = \mathfrak p _ {v} , v \notin S, } \\ 0 & {\textrm{ otherwise } , } \\ \end{array} \right .$$

and extending $\chi$ multiplicatively. These functions are called Hecke characters or Grössencharakters. Given such a character, the Hecke zeta-function of $\chi$ is defined by

$$\zeta ( s , \chi ) = \prod _ { \mathfrak p } \left ( 1 - \frac{\chi ( \mathfrak p ) }{N( \mathfrak p ) ^ {s} } \right ) ^ {-1} = \ \sum _ { \mathfrak a } \frac{\chi ( \mathfrak a ) }{N( \mathfrak a ) ^ {s} } ,$$

where $N$ is the absolute norm $G( K) \rightarrow G( \mathbf Q )$. The function $\zeta ( s, \chi )$ is also called $L$-series, Dirichlet $L$-series (when $\chi$ is a Dirichlet character) or Hecke $L$-function with Grössencharakter; it is also denoted by $L( s, \chi )$. If $\chi \equiv 1$ one obtains the Dedekind $\zeta$-function. For Dirichlet $L$-series the generalized Riemann hypothesis states that $L ( s, \chi ) \neq 0$ if $\mathop{\rm Re} ( s) > 1/2$.

How to Cite This Entry:
Riemann hypothesis, generalized. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_hypothesis,_generalized&oldid=52259
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article