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:
-functions are meromorphic functions on the complex plane, which behave like analytic continuations of traces of powers
of suitable elliptic differential operators (in physics these are regularized traces of Feynman propagators leading to expressions for vacuum amplitudes), which means that for sufficiently nice such these are analytic continuations in of sums of the form
where the summation is over the eigenvalues of .
Indeed, such zeta functions of elliptic differential operators constitutes one class of examples of zeta functions. Of particular interest is the case where 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 on the complex plane such that
for 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 on the complex plane;
and such that the result satisfies a functional equation which says that the product of with some correcion functions satisfies .
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).
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
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 , Proc. Japan Acad. Ser. A Math. Sci. 81:10 (2005) 180-184 euclid
Bruno Kahn, Fonctions zêta et de variétés et de motifs, arXiv:1512.09250.
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
Sergey Galkin?, Evgeny Shinder?, On a zeta-function of a dg-category, arXiv:1506.05831.
Last revised on May 16, 2023 at 12:32:35. See the history of this page for a list of all contributions to it.