Dedekind zeta function: Difference between revisions
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In [[mathematics]], to each [[algebraic number field]] ''k'', there is associated an important function called the '''Dedekind zeta function'''. This function is a generalization of the [[Riemann zeta function]], and its definition is similar. | In [[mathematics]], to each [[algebraic number field]] ''k'', there is associated an important function called the '''Dedekind zeta function'''. This function is a generalization of the [[Riemann zeta function]], and its definition is similar. Its importance stems from the fact that it encodes all of the important ''arithmetic'' information about the field in an ''analytic manner''. As such, important results about the arithmetic in the field can be obtained from the associated zeta function through analytic methods. | ||
== Definition == | == Definition == | ||
If ''k'' is an algebraic number field, the Dedekind zeta function of the field is a [[meromorphic function]], defined for complex numbers ''s'' with real part satisfying <math> | If ''k'' is an algebraic number field, the Dedekind zeta function of the field is a [[meromorphic function]], defined for complex numbers ''s'' with real part satisfying <math> \Re (s) > 1 </math> by the [[Dirichlet series]] | ||
: <math> \zeta_{k} (s) = \sum_{\mathfrak{a}} \frac{1}{\mathfrak{N} (\mathfrak{a})^{s}}, </math> | : <math> \zeta_{k} (s) = \sum_{\mathfrak{a}} \frac{1}{\mathfrak{N} (\mathfrak{a})^{s}}, </math> | ||
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where the sum extends over the set of integral [[ideals]] of ''k'', and <math> \mathfrak{N} (\mathfrak{a}) </math> denotes their [[absolute norm]]. | where the sum extends over the set of integral [[ideals]] of ''k'', and <math> \mathfrak{N} (\mathfrak{a}) </math> denotes their [[absolute norm]]. | ||
This series is [[absolutely convergent]] on [[compact subsets]] of the half-plane <math> \Re (s) > 1 </math>. It thus defines a holomorphic function on this half-plane, and this can be extended by analytic continuation to a meromorphic function on the whole complex plane. It is holomorphic everywhere except at ''s'' = 1, where it has a simple pole. The zeta function <math> \zeta_{\mathbb{Q}} (s) </math> associated to the field of rational numbers is just the Riemann zeta function. | This series is [[absolutely convergent]] on [[compact subsets]] of the half-plane <math> \Re (s) > 1 </math>. It thus defines a holomorphic function on this half-plane, and this can be extended by analytic continuation to a meromorphic function on the whole complex plane. It is holomorphic everywhere except at ''s'' = 1, where it has a simple pole. The zeta function <math> \zeta_{\mathbb{Q}} (s) </math> associated to the field of rational numbers is just the Riemann zeta function. | ||
==Properties== | |||
The Dedekind zeta function has an ''[[Euler product]]'': | |||
== | :<math>\zeta_{k} (s) = \sum_{\mathfrak{a}} \frac{1}{\mathfrak{N} (\mathfrak{a})^{s}} | ||
= \prod_{\mathfrak{p}} \left(1-\mathfrak{N} (\mathfrak{p})^{-s}\right)^{-1} | |||
, </math> | |||
where <math>\mathfrak{p}</math> runs over [[prime ideal]]s of the ring of integers, which formally expresses the unique factorisation of ideals into prime ideals. | |||
Just like the Riemann zeta function, each Dedekind zeta function possesses a functional equation. Furthermore, the [[analytic class number formula]], which appears trivial in the context of the Riemann zeta function, expresses the residue of the pole at ''s'' =   1 in terms of the important arithmetic invariants of the field. | Just like the Riemann zeta function, each Dedekind zeta function possesses a functional equation. Furthermore, the [[analytic class number formula]], which appears trivial in the context of the Riemann zeta function, expresses the residue of the pole at ''s'' = 1 in terms of the important arithmetic invariants of the field.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 12:01, 5 August 2024
In mathematics, to each algebraic number field k, there is associated an important function called the Dedekind zeta function. This function is a generalization of the Riemann zeta function, and its definition is similar. Its importance stems from the fact that it encodes all of the important arithmetic information about the field in an analytic manner. As such, important results about the arithmetic in the field can be obtained from the associated zeta function through analytic methods.
Definition
If k is an algebraic number field, the Dedekind zeta function of the field is a meromorphic function, defined for complex numbers s with real part satisfying by the Dirichlet series
where the sum extends over the set of integral ideals of k, and denotes their absolute norm.
This series is absolutely convergent on compact subsets of the half-plane . It thus defines a holomorphic function on this half-plane, and this can be extended by analytic continuation to a meromorphic function on the whole complex plane. It is holomorphic everywhere except at s = 1, where it has a simple pole. The zeta function associated to the field of rational numbers is just the Riemann zeta function.
Properties
The Dedekind zeta function has an Euler product:
where runs over prime ideals of the ring of integers, which formally expresses the unique factorisation of ideals into prime ideals.
Just like the Riemann zeta function, each Dedekind zeta function possesses a functional equation. Furthermore, the analytic class number formula, which appears trivial in the context of the Riemann zeta function, expresses the residue of the pole at s = 1 in terms of the important arithmetic invariants of the field.