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.

Comments

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 $\mathfrak P$,

$$\chi ( \mathfrak P ) = \left \{ and extending $ \chi $ multiplicatively. These functions are called Hecke characters or Grössencharakters. Given such a character, the Hecke zeta-function of $ \chi $ is defined by$$ \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 $\mathop{\rm Re} ( s) > 1/2$.

References

[a1]	H. Heilbronn, "Zeta-functions and -functions" J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) pp. 204–230
[a2]	W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , Springer & PWN (1990) pp. Chapt. 7, §1