In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q).
It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.
The Dedekind zeta function is named for Richard Dedekind who introduced it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.
Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index [OK : I] of I in OK or equivalently the cardinality of the quotient ring OK / I).
This sum converges absolutely for all complex numbers s with real part Re(s) > 1.
In the case K = Q, this definition reduces to that of the Riemann zeta function.
This is the expression in analytic terms of the uniqueness of prime factorization of ideals in
The residue at that pole is given by the analytic class number formula and is made up of important arithmetic data involving invariants of the unit group and class group of K. The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let ΔK denote the discriminant of K, let r1 (resp.
Then, the functions satisfy the functional equation Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h(K) of K, the regulator R(K) of K, the number w(K) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of OK and the leading term is given by It follows from the functional equation that
In the totally real case, Carl Ludwig Siegel showed that ζK(s) is a non-zero rational number at negative odd integers.
Stephen Lichtenbaum conjectured specific values for these rational numbers in terms of the algebraic K-theory of K. For the case in which K is an abelian extension of Q, its Dedekind zeta function can be written as a product of Dirichlet L-functions.
That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet L-function is an analytic formulation of the quadratic reciprocity law of Gauss.
): for general extensions the result would follow from the Artin conjecture for L-functions.
[2] Additionally, ζK(s) is the Hasse–Weil zeta function of Spec OK[3] and the motivic L-function of the motive coming from the cohomology of Spec K.[4] Two fields are called arithmetically equivalent if they have the same Dedekind zeta function.
Wieb Bosma and Bart de Smit (2002) used Gassmann triples to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent.