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A statement about the non-trivial zeros of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819401.png" />-functions (cf. [[Dirichlet L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819402.png" />-function]]), Dedekind zeta-functions (cf. [[Zeta-function|Zeta-function]]) and several other similar functions, similar to the Riemann hypothesis (cf. [[Riemann hypotheses|Riemann hypotheses]]) on the non-trivial zeros of the Riemann zeta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819403.png" />. In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Riemann hypothesis.
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A statement about the non-trivial zeros of Dirichlet  $  L $-functions (cf. [[Dirichlet L-function|Dirichlet  $  L $-function]]), Dedekind zeta-functions (cf. [[Zeta-function|Zeta-function]]) and several other similar functions, similar to the Riemann hypothesis (cf. [[Riemann hypotheses|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====
 
====Comments====
For Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819404.png" />-functions it is not even known whether there exist real zeros in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819405.png" /> (Siegel zeros). This is important in connection with the class number of quadratic fields (see also [[Quadratic field|Quadratic field]]; [[Siegel theorem|Siegel theorem]]).
+
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|Quadratic field]]; [[Siegel theorem|Siegel theorem]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819407.png" /> be an algebraic number field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819408.png" /> the group of fractional ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r0819409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194010.png" /> its idèle class group (cf. [[Idèle|Idèle]]; [[Fractional ideal|Fractional ideal]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194011.png" /> be a quasi-character on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194012.png" />, i.e. a continuous homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194013.png" /> into the group of non-zero complex numbers. Then for an idèle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194014.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194015.png" />, where for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194017.png" /> is a quasi-character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194018.png" /> which is equal to unity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194019.png" />, the units of the local completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194020.png" />, for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194021.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194022.png" /> be a finite subset of the valuations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194023.png" /> including the Archimedian ones, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194024.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194025.png" /> can now be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194026.png" /> by setting for all prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194027.png" />,
+
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|Idèle]]; [[Fractional ideal|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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194028.png" /></td> </tr></table>
+
$$
 +
\chi ( \mathfrak P )  = \left \{
  
and extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194029.png" /> multiplicatively. These functions are called Hecke characters or Grössencharakters. Given such a character, the Hecke zeta-function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194030.png" /> is defined by
+
\begin{array}{cl}
 +
{X _ {v} ( \mathfrak p v) }  & {\textrm{ if }  \mathfrak P = \mathfrak p _ {v} , v \notin S, }  \\
 +
0 & {\textrm{ otherwise } , }  \\
 +
\end{array}
 +
\right .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194031.png" /></td> </tr></table>
+
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
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194032.png" /> is the absolute norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194033.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194034.png" /> is also called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194036.png" />-series, Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194038.png" />-series (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194039.png" /> is a Dirichlet character) or Hecke <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194041.png" />-function with Grössencharakter; it is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194043.png" /> one obtains the Dedekind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194045.png" />-function. For Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194046.png" />-series the generalized Riemann hypothesis states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194047.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194048.png" />.
+
$$
 +
\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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Heilbronn,  "Zeta-functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194049.png" />-functions"  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)  pp. 204–230</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Narkiewicz,  "Elementary and analytic theory of algebraic numbers" , Springer &amp; PWN  (1990)  pp. Chapt. 7, §1</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Heilbronn,  "Zeta-functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194049.png" />-functions"  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)  pp. 204–230</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Narkiewicz,  "Elementary and analytic theory of algebraic numbers" , Springer &amp; PWN  (1990)  pp. Chapt. 7, §1</TD></TR></table>

Latest revision as of 10:52, 21 March 2022


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
How to Cite This Entry:
Riemann hypothesis, generalized. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_hypothesis,_generalized&oldid=36173
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article