Difference between revisions of "Riemann hypothesis, generalized"

Revision as of 14:53, 7 June 2020

A statement about the non-trivial zeros of Dirichlet -functions (cf. Dirichlet -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 . In the case of Dirichlet zeta-functions the generalized Riemann hypothesis is called the extended Riemann hypothesis.

Comments

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

Let be an algebraic number field, the group of fractional ideals of and its idèle class group (cf. Idèle; Fractional ideal). Let be a quasi-character on , i.e. a continuous homomorphism of into the group of non-zero complex numbers. Then for an idèle one has , where for each , is a quasi-character of which is equal to unity on , the units of the local completion , for almost-all . Let be a finite subset of the valuations on including the Archimedian ones, . A function can now be defined on by setting for all prime ideals ,

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

where is the absolute norm . The function is also called -series, Dirichlet -series (when is a Dirichlet character) or Hecke -function with Grössencharakter; it is also denoted by . If one obtains the Dedekind -function. For Dirichlet -series the generalized Riemann hypothesis states that if .

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=49403

This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

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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.
-r0819401.png
-$#A+1 = 43 n = 1
-$#C+1 = 43 : ~/encyclopedia/old_files/data/R081/R.0801940 Riemann hypothesis, generalized
-Automatically converted into TeX, above some diagnostics.
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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====
-For Dirichlet  $  L $-
+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]]).
-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 <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" />,
-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 $,
-$$
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194028.png" /></td> </tr></table>
-\chi ( \mathfrak P )  =  \left \{
-and extending  $  \chi $
+and extending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194029.png" /> multiplicatively. These functions are called Hecke characters or Grössencharakters. Given such a character, the Hecke zeta-function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081940/r08194030.png" /> is defined by
-multiplicatively. These functions are called Hecke characters or Grössencharakters. Given such a character, the Hecke zeta-function of  $  \chi $
-is defined by
-$$
+<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>
-\zeta ( s , \chi )  =  \prod _ { \mathfrak p }
-\left ( 1 -
-\frac{\chi ( \mathfrak p ) }{N( \mathfrak p )  ^ {s} }
-\right )  ^ {-} 1  = \
+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" />.
-\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====
 <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>

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Difference between revisions of "Riemann hypothesis, generalized"

Revision as of 14:53, 7 June 2020

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References