The concept of zeta function originates in number theory, but to get an idea of what they “really are” it is helpful to proceed anachronistically:
$\zeta$-functions are meromorphic functions $s \mapsto \zeta(s)$ on the complex plane, which behave like like analytic continuations of traces of powers
of suitable elliptic differential operators $H$ (in physics these are regularized traces of Feynman propagators leading to expressions for vacuum amplitudes), which means that for sufficiently nice such $H$ these are analytic continuations in $s$ of sums of the form
where the summation is over the eigenvalues $\lambda$ of $H$.
Indeed, such zeta functions of elliptic differential operators constitutes one class of examples of zeta functions. Of particular interest is the case where $H$ is a Laplace operator of a hyperbolic manifold and in particular on a hyperbolic Riemann surface, for that case one obtains the zeta function of a Riemann surface, in particular the Selberg zeta function.
In modern language one also speaks of L-functions. Where a zeta function of some space is like the Feynman propagator of the canonical Laplace operator of that space, an L-function is defined from an extra “twisting” information such as that of a flat bundle/local system of coefficients on the space (and is hence like the Feynman propagator of the corresponding twisted/coupled Laplace operator). The major properties satisfied by anything that qualifies as a zeta function or L-functions are: these are meromorphic functions $s \mapsto L(s)$ on the complex plane such that
for $\Re(s) \gt 1$ they have a converging series expansion of the above form, and/or a multiplicative series expression, the Euler product;
such that analytic continuation of the series expression exists to a meromorphic function $L(-)$ on the complex plane;
and such that the result satisfies a functional equation which says that the product $\hat L$ of $L$ with some correcion functions satisfies $\hat L(1-s) = \hat L(s)$.
Proceeding from the above class of examples in complex analytic geometry one may wonder if there are analogs also in arithmetic geometry. Indeed, by the function field analogy there are. All the way down “on Spec(Z)” the analog of the Selberg zeta function is the Riemann zeta function, which historically is the first of all zeta functions, defined by analytic continuation of the series
The Riemann hypothesis conjectures a characterization of the roots of this zeta function and is regarded as one of the outstanding problems in mathematics. It has evident analogs for all other zeta functions (for some of which it has been proven).
More generally, over arithmetic curves which are spectra of rings of integers of more general number fields, the Riemann zeta function has generalization to the Artin L-functions defined intrinsically in terms of characteristic polynomials of Galois representations. When the Galois representation is 1-dimensional, then the Artin L-function may be expressed (by “Artin reciprocity”) in terms of “more arithmetic” data by Dirichlet L-functions and Hecke L-functions. When the Galois representation is higher dimensional, then the Langlands correspondence conjecture asserts that the Artin L-function may be expressed “arithmetically” as the automorphic L-function of an automorphic form.
Similarly on arithmetic curves given by function fields there is the Goss zeta function and in higher dimensional arithmetic geometry the Weil zeta function, famous from the Weil conjectures. The
When interpreting the Frobenius morphisms that appear in the Artin L-functions geometrically as flows (as discussed at Borger’s arithmetic geometry – Motivation) then this induces an evident analog of zeta function of a dynamical system. This in turn has strong analogies with Alexander polynomials in knot theory (see at arithmetic topology).
context/function field analogy | theta function $\theta$ | zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$) | L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$) | eta function $\eta$ | special values of L-functions |
---|---|---|---|---|---|
physics/2d CFT | partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$ | analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$ | analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$ | analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$ | regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$ |
Riemannian geometry (analysis) | zeta function of an elliptic differential operator | zeta function of an elliptic differential operator | eta function of a self-adjoint operator | functional determinant, analytic torsion | |
complex analytic geometry | section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$ | zeta function of a Riemann surface | Selberg zeta function | Dedekind eta function | |
arithmetic geometry for a function field | Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry) | ||||
arithmetic geometry for a number field | Hecke theta function, automorphic form | Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation) | Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps) | class number $\cdot$ regulator | |
arithmetic geometry for $\mathbb{Q}$ | Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character) | Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$) | Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function |
One way to understand the plethora of different zeta functions is to see them as the incarnation of the same general concept in different flavors of geometry. This is expressed at least in parts by the
number fields (“function fields of curves over F1”) | function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves) | Riemann surfaces/complex curves | |
---|---|---|---|
affine and projective line | |||
$\mathbb{Z}$ (integers) | $\mathbb{F}_q[z]$ (polynomials, function algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$) | $\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane) | |
$\mathbb{Q}$ (rational numbers) | $\mathbb{F}_q(z)$ (rational functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (place at infinity) | $\infty$ | ||
$Spec(\mathbb{Z})$ (Spec(Z)) | $\mathbb{A}^1_{\mathbb{F}_q}$ (affine line) | complex plane | |
$Spec(\mathbb{Z}) \cup place_{\infty}$ | $\mathbb{P}_{\mathbb{F}_q}$ (projective line) | Riemann sphere | |
$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient) | $\frac{\partial}{\partial z}$ (coordinate derivation) | “ | |
genus of the rational numbers = 0 | genus of the Riemann sphere = 0 | ||
formal neighbourhoods | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$) | $\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$) | |
$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$) | formal disks in $X$ | ||
$\mathbb{Q}_p$ (p-adic numbers) | $\mathbb{F}_q((z-x))$ (Laurent series around $x$) | $\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$) | |
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles) | $\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks) | |
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles) | $\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$ | |
theta functions | |||
Jacobi theta function | |||
zeta functions | |||
Riemann zeta function | Goss zeta function | ||
branched covering curves | |||
$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension) | $K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$ | $K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$) | |
$\mathcal{O}_K$ (ring of integers) | $\mathcal{O}_{\Sigma}$ (structure sheaf) | ||
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places) | $\Sigma$ (arithmetic curve) | $\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere) | |
$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure) | $\frac{\partial}{\partial z}$ | “ | |
genus of a number field | genus of an algebraic curve | genus of a surface | |
formal neighbourhoods | |||
$v$ prime ideal in ring of integers $\mathcal{O}_K$ | $x \in \Sigma$ | $x \in \Sigma$ | |
$K_v$ (formal completion at $v$) | $\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$) | ||
$\mathcal{O}_{K_v}$ (ring of integers of formal completion) | $\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$) | ||
$\mathbb{A}_K$ (ring of adeles) | $\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$) | ||
$\mathcal{O}$ | $\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$) | ||
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles) | $\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$ | ||
Galois theory | |||
Galois group | “ | $\pi_1(\Sigma)$ fundamental group | |
Galois representation | “ | flat connection (“local system”) on $\Sigma$ | |
class field theory | |||
class field theory | “ | geometric class field theory | |
Hilbert reciprocity law | Artin reciprocity law | Weil reciprocity law | |
$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group) | “ | ||
$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$ | “ | $Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem) | |
non-abelian class field theory and automorphy | |||
number field Langlands correspondence | function field Langlands correspondence | geometric Langlands correspondence | |
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations) | “ | $Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem) | |
Tamagawa-Weil for number fields | Tamagawa-Weil for function fields | ||
theta functions | |||
Hecke theta function | functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$ | ||
zeta functions | |||
Dedekind zeta function | Weil zeta function | zeta function of a Riemann surface/of the Laplace operator on $\Sigma$ | |
higher dimensional spaces | |||
zeta functions | Hasse-Weil zeta function |
multiple zeta values, motivic multiple zeta values, motivic integration, motive
there are attempts to understand the Riemann zeta function as the spectrum of a Hamiltonian of a quantum mechanical system. See at Riemann hypothesis and physics.
A useful survey of the zoo of zeta functions is in
Further general review includes
E. Kowalski, first part of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)
Alain Connes, Matilde Marcolli, chapter II of Noncommutative Geometry, Quantum Fields and Motives
Discussion in the more general context of higher dimensional arithmetic geometry is in
Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228:4 (1995) 121–163, and preprint MPIM1992-50 pdf
Nobushige Kurokawa, Zeta functions over $F_1$, Proc. Japan Acad. Ser. A Math. Sci. 81:10 (2005) 180-184 euclid
M. Larsen, V. A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3, 1 (2003) 85–95; Rationality criteria for motivic zeta functions, Compos. Math. 140:6 (2004) 1537–1560
Vladimir Guletskii, Zeta functions in triangulated categories, Mathematical Notes 87, 3 (2010) 369–381, math/0605040
M. Kontsevich, Notes on motives in finite characteristics, math.AG/0702206