Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζ_K(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 ζ_K(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.[1]

Definition and basic properties

Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series

${\displaystyle \zeta _{K}(s)=\sum _{I\subseteq {\mathcal {O}}_{K}}{\frac {1}{(N_{K/\mathbf {Q} }(I))^{s}}}}$

where I ranges through the non-zero ideals of the ring of integers O_K of K and N_K/Q(I) denotes the absolute norm of I (which is equal to both the index [O_K : I] of I in O_K or equivalently the cardinality of quotient ring O_K / 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.

Euler product

The Dedekind zeta function of ${\displaystyle K}$ has an Euler product which is a product over all the non-zero prime ideals ${\displaystyle {\mathfrak {p}}}$ of ${\displaystyle {\mathcal {O}}_{K}}$

${\displaystyle \zeta _{K}(s)=\prod _{{\mathfrak {p}}\subseteq {\mathcal {O}}_{K}}{\frac {1}{1-N_{K/\mathbf {Q} }({\mathfrak {p}})^{-s}}},{\text{ for Re}}(s)>1.}$

This is the expression in analytic terms of the uniqueness of prime factorization of ideals in ${\displaystyle {\mathcal {O}}_{K}}$. For ${\displaystyle \mathrm {Re} (s)>1,\ \zeta _{K}(s)}$ is non-zero.

Analytic continuation and functional equation

Erich Hecke first proved that ζ_K(s) has an analytic continuation to a meromorphic function that is analytic at all points of the complex plane except for one simple pole at s = 1. 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 Δ_K denote the discriminant of K, let r₁ (resp. r₂) denote the number of real places (resp. complex places) of K, and let

${\displaystyle \Gamma _{\mathbf {R} }(s)=\pi ^{-s/2}\Gamma (s/2)}$

and

${\displaystyle \Gamma _{\mathbf {C} }(s)=(2\pi )^{-s}\Gamma (s)}$

where Γ(s) is the gamma function. Then, the functions

${\displaystyle \Lambda _{K}(s)=\left|\Delta _{K}\right|^{s/2}\Gamma _{\mathbf {R} }(s)^{r_{1}}\Gamma _{\mathbf {C} }(s)^{r_{2}}\zeta _{K}(s)\qquad \Xi _{K}(s)={\tfrac {1}{2}}(s^{2}+{\tfrac {1}{4}})\Lambda _{K}({\tfrac {1}{2}}+is)}$

satisfy the functional equation

${\displaystyle \Lambda _{K}(s)=\Lambda _{K}(1-s).\qquad \Xi _{K}(-s)=\Xi _{K}(s)\;}$

Special values

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 O_K and the leading term is given by

${\displaystyle \lim _{s\rightarrow 0}s^{-r}\zeta _{K}(s)=-{\frac {h(K)R(K)}{w(K)}}.}$

It follows from the functional equation that ${\displaystyle r=r_{1}+r_{2}-1}$. Combining the functional equation and the fact that Γ(s) is infinite at all integers less than or equal to zero yields that ζ_K(s) vanishes at all negative even integers. It even vanishes at all negative odd integers unless K is totally real (i.e. r₂ = 0; e.g. Q or a real quadratic field). In the totally real case, Carl Ludwig Siegel showed that ζ_K(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.

Relations to other L-functions

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. For example, when K is a quadratic field this shows that the ratio

${\displaystyle {\frac {\zeta _{K}(s)}{\zeta _{\mathbf {Q} }(s)}}}$

is the L-function L(s, χ), where χ is a Jacobi symbol used as Dirichlet character. 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.

In general, if K is a Galois extension of Q with Galois group G, its Dedekind zeta function is the Artin L-function of the regular representation of G and hence has a factorization in terms of Artin L-functions of irreducible Artin representations of G.

The relation with Artin L-functions shows that if L/K is a Galois extension then ${\displaystyle {\frac {\zeta _{L}(s)}{\zeta _{K}(s)}}}$ is holomorphic (${\displaystyle \zeta _{K}(s)}$ "divides" ${\displaystyle \zeta _{L}(s)}$): for general extensions the result would follow from the Artin conjecture for L-functions.[2]

Additionally, ζ_K(s) is the Hasse–Weil zeta function of Spec O_K[3] and the motivic L-function of the motive coming from the cohomology of Spec K.[4]

Arithmetically equivalent fields

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. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.

Perlis (1977) 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 in the two fields, i.e., if ${\displaystyle {\mathfrak {p}}_{i}}$ are the prime ideals in K lying over p, then the tuples ${\displaystyle (\dim _{\mathbf {Z} /p}{\mathcal {O}}_{K}/{\mathfrak {p}}_{i})}$ need to be the same for K and for L for almost all p.

Notes

1. Narkiewicz 2004, §7.4.1
2. Martinet (1977) p.19
3. Deninger 1994, §1
4. Flach 2004, §1.1

References

• Bosma, Wieb; de Smit, Bart (2002), "On arithmetically equivalent number fields of small degree", in Kohel, David R.; Fieker, Claus (eds.), Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci., vol. 2369, Berlin, New York: Springer-Verlag, pp. 67–79, doi:10.1007/3-540-45455-1_6, ISBN 978-3-540-43863-2, MR 2041074
• Section 10.5.1 of Cohen, Henri (2007), Number theory, Volume II: Analytic and modern tools, Graduate Texts in Mathematics, vol. 240, New York: Springer, doi:10.1007/978-0-387-49894-2, ISBN 978-0-387-49893-5, MR 2312338
• Deninger, Christopher (1994), "L-functions of mixed motives", in Jannsen, Uwe; Kleiman, Steven; Serre, Jean-Pierre (eds.), Motives, Part 1, Proceedings of Symposia in Pure Mathematics, vol. 55, American Mathematical Society, pp. 517–525, ISBN 978-0-8218-1635-6
• Flach, Mathias (2004), "The equivariant Tamagawa number conjecture: a survey", in Burns, David; Popescu, Christian; Sands, Jonathan; et al. (eds.), Stark's conjectures: recent work and new directions (PDF), Contemporary Mathematics, vol. 358, American Mathematical Society, pp. 79–125, ISBN 978-0-8218-3480-0
• Martinet, J. (1977), "Character theory and Artin L-functions", in Fröhlich, A. (ed.), Algebraic Number Fields, Proc. Symp. London Math. Soc., Univ. Durham 1975, Academic Press, pp. 1–87, ISBN 0-12-268960-7, Zbl 0359.12015
• Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: Springer-Verlag, Chapter 7, ISBN 978-3-540-21902-6, MR 2078267
• Perlis, Robert (1977), "On the equation ${\displaystyle \zeta _{K}(s)=\zeta _{K'}(s)}$", Journal of Number Theory, 9 (3): 342–360, doi:10.1016/0022-314X(77)90070-1