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 $,

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

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 $.

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