In mathematics, the Dedekind zeta function of an algebraic number field K, is an analytic function that represents information about the ideals in the corresponding number ring, generalizing how the Riemann zeta function represents information about the factorization of integers.

The Dedekind zeta function generalizes many properties of the Riemann zeta function: it can be defined as a Dirichlet series; it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole; it has an Euler product expansion; and it satisfies a functional equation. Values of Dedekind zeta functions encode important arithmetic data of K.

The Dedekind zeta function is named for Richard Dedekind, who introduced it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.

Motivation

Unique factorization of non-zero elements into powers of prime elements, which is a fundamental property for usual (rational) integer numbers, generally fails for the ring of integers of an arbitrary number field, which is not necessarily a principal ideal domain (PID) (although Gaussian integers and Eisenstein integers are PIDs).

However, since the ring of integers is a Dedekind ring, uniqueness does hold for non-zero idealss factored into powers of prime ideals. Thus for general number rings, prime ideals rather than prime elements are the main objects of interest. To describe the distribution of prime ideals among all ideals requires a measurement of an ideal's "size". The natural measure is the absolute norm, defined as the cardinality of the quotient ring:

For rational integers, this is equal to the non-negative integer generating the ideal.

Definition

be an algebraic number field, with a ring of integers is the analytic function defined by the series:

); over the rest of the complex plane, it is defined as the analytic continuation of this series.

Basic properties

Dirichlet series

, we can rewrite series in the domain of absolute convergence as:

Using Minkowski's bound and summing over all ideal classes one can show the growth bound:

and defines a holomorphic function in this domain.

Euler product

For every number field, its ring of integers is a Dedekind domain, hence every ideal can be uniquely factored into a product of prime ideals. The norm function is multiplicative with respect to multiplication of ideals, which implies that the Dedekind zeta function has an Euler product:

in this half-plane.

Analytic continuation

Erich Hecke has an analytic continuation to a meromorphic function that is analytic at all points of the complex plane except for one simple pole at . The residue at that pole is given by the class number formula (see below), which combines important arithmetic data involving invariants of the unit group and class group.

Functional equation

The Dedekind zeta function satisfies a functional equation, generalizing the equation satisfied by the Riemann zeta function. The functional equation involves important invariants of number field. Let:

• denote the discriminant,

• denote the number of real places,

• .

In terms of the gamma function, define real and complex gamma factors as:

Then, the function:

satisfies the functional equation:

The functional equation for the Dedekind zeta function implies a set of trivial zeroes which cancel poles of the gamma factors in the equation; while nontrivial zeroes.

function defined from the Riemann zeta function, one can define:

1. ,

, producing an entire function, and moves the critical line to the real line. The functional equation simplifies to:

Special values

As for the Riemann zeta function, the values of the Dedekind zeta function at integers (or related quantities like the residue, the multiplicities of zeroes, or the leading coefficient in the Taylor expansion at zero) may encode important arithmetic data of the field K, at least conjecturally. Let:

• ,

• ,

• ,

• ,

• ,

• .

Dedekind's class number formula to this data:

, which equals the rank, the leading term at this point is:

1. .

is a totally real number field, in which case Siegel is a non-zero rational number.

Arithmetically equivalent fields

Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. Such pairs are useful as counterexamples to show which arithmetic invariants cannot be determined by any features of the Dedekind zeta function.

showed that two number fields K and L are arithmetically equivalent if and only if all but finitely many prime numbers p have the same inertia degrees are the prime ideals in K lying over p need to be the same for K and for L for almost all p.

used Gassmann triples.

Relations to other L-functions and zeta functions

Special cases

, the Dedekind zeta function is equal to the Riemann zeta function, the classical prototype for all zeta functions and L-functions.

More general zeta functions

The Dedekind zeta function is a special case of an arithmetic zeta function and of a Hasse–Weil zeta function for the scheme. It is also the motivic L-function of the motive coming from the cohomology.

Artin L-functions

Although Artin L-functions and the trivial representation of its Galois group, the resulting Artin L-function is:

is a finite Galois extension, then the Dedekind zeta function of the larger field is the Artin L-function for the regular representation, and has a factorization into L-functions of irreducible representations of this group:

Without the assumption that the extension is Galois, the formula becomes more complicated, but is also possible to obtain a similar factorization using the normal closure of the larger field and induced representations. Let:

• ,

• ,

• ,

• be the inner product on characters of representations,

• ,

• ,

• .

Then the factorization of the Dedekind zeta function is as follows:

Hecke L-functions

is an abelian extension, Artin reciprocity allows the factorization to be described in terms of Hecke L-functions:

where the index runs over primitive Hecke characters corresponding to irreducible representations of the abelian Galois group.

Dirichlet L-functions

is an abelian extension, the Hecke characters become Dirichlet characters and Hecke L-functions become Dirichlet L-functions.

In the simple example of a quadratic field K, an abelian extension of the rational numbers, this becomes:

is a Jacobi symbol used as a Dirichlet character. This fact—that the zeta function of a quadratic field is a product of the Riemann zeta function and the Dirichlet L-function with Jacobi symbol—is an analytic formulation of the quadratic reciprocity law.

Conjectures

Dedekind conjecture

Dedekind conjectured that for every number field function,

is an entire function of number field, the quotient

is an entire function. For abelian extensions, the Dedekind conjecture follows from factorization into Hecke L-functions and fact that Hecke L-functions for nontrivial characters are entire. For general Galois extensions, this follows from the Aramata–Brauer theorem. For extensions which are contained in solvable extensions, it was proven independently by and .

The general case is still open, but follows directly from more general conjectures like the Artin conjecture or the Selberg orthonormality conjecture.

Extended Riemann hypothesis

. The extended Riemann hypothesis (ERH) says that all nontrivial zeros of Dedekind zeta functions lie on the critical line, generalizing the classical Riemann hypothesis, an abelian extension of the rational numbers.

Many results in analytic number theory and algebraic number theory follow from this conjecture.

Values at negative integers

There are attempts to generalize the relation between Bernoulli numbers and values of the Riemann zeta function at negative odd integers:

are nonzero rational numbers. Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K-theory of K.

Notes

References

• Section 10.5.1 of