Difference between revisions of "Riemann hypothesis, generalized"
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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 | + | 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. |
Revision as of 21:18, 9 January 2015
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 ![]() |
[a2] | W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , Springer & PWN (1990) pp. Chapt. 7, §1 |
Riemann hypothesis, generalized. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_hypothesis,_generalized&oldid=36173