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Difference between revisions of "Riemann hypothesis, generalized"

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A statement about the non-trivial zeros of Dirichlet  $  L $-
+
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) $.  
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.
 
In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Riemann hypothesis.
  
 
====Comments====
 
====Comments====
For Dirichlet  $  L $-
+
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]]).
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  $  K $
 
Let  $  K $
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\frac{\chi ( \mathfrak p ) }{N( \mathfrak p )  ^ {s} }
 
\frac{\chi ( \mathfrak p ) }{N( \mathfrak p )  ^ {s} }
  
\right )  ^ {-} 1 = \  
+
\right )  ^ {-1}  = \  
 
\sum _ { \mathfrak a }
 
\sum _ { \mathfrak a }
  
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is the absolute norm  $  G( K) \rightarrow G( \mathbf Q ) $.  
 
is the absolute norm  $  G( K) \rightarrow G( \mathbf Q ) $.  
 
The function  $  \zeta ( s, \chi ) $
 
The function  $  \zeta ( s, \chi ) $
is also called  $  L $-
+
is also called  $  L $-series, Dirichlet  $  L $-series (when  $  \chi $
series, Dirichlet  $  L $-
+
is a Dirichlet character) or Hecke  $  L $-function with Grössencharakter; it is also denoted by  $  L( s, \chi ) $.  
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 $
 
If  $  \chi \equiv 1 $
one obtains the Dedekind  $  \zeta $-
+
one obtains the Dedekind  $  \zeta $-function. For Dirichlet  $  L $-series the generalized Riemann hypothesis states that  $  L ( s, \chi ) \neq 0 $
function. For Dirichlet  $  L $-
 
series the generalized Riemann hypothesis states that  $  L ( s, \chi ) \neq 0 $
 
 
if  $  \mathop{\rm Re} ( s) > 1/2 $.
 
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=49562
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article