a directory of all known zeta functions
[This page is under continual construction! Any contributions would be welcome.]
"Over the years striking analogies have been observed between the Riemann zetafunction and
other zeta or Lfunctions. While these functions are seemingly independent of each other, there
is growing evidence that they are all somehow connected in a way that we do not fully understand. In
any event, trying to understand, or at least classify, all of the objects which we believe satisfy the
Riemann hypothesis is a reasonable thing to do."
J. Brian Conrey, "The Riemann Hypothesis", Notices of the AMS (March, 2003) p.347
"In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas
underlying the theory...
Whenever entities are counted with some mathematical structure on them it is likely that a zeta function can be set up and often enough it will extend to a meromorphic function. Zeta functions show up in all areas of mathematics and they encode properties of the counted objects which are well hidden and hard to come by otherwise. They easily give fuel for bold new conjectures and thus drive on mathematical research. It is a fairly safe assertion to say that zeta functions of various kinds will stay in the focus of mathematical attention for times to come. "
A. Deitmar, "Panorama of zeta functions" (preprint 10/02)
"Some decades ago I made  somewhat in jest  the suggestion that one should get accepted a nonproliferation treaty of zeta functions. There was becoming such an overwhelming variety of these objects."
Atle Selberg, quoted in K. Sabbagh, Dr. Riemann's Zeros (Atlantic, 2002)
"In concentrating exclusively on the study of the [Riemann] zeta function and its
relation to the prime number theorem, this book ignores one of the most fruitful areas of
development of Riemann's work, namely, number theory. The use of functions like the zeta
function in number theory was a major feature of the work of Dirichlet
both in his Lseries and in his formula for the class number of
a quadratic number field  many years before Riemann's paper appeared,
and the use of such functions has been a prominent theme in number theory
ever since. Riemann's contributions in this area were primarily
functiontheoretic, not numbertheoretic, and consisted of focussing
attention on the functions as functions of a complex variable,
on the possibility of their satisfying a functional equation under s
<> 1  s, and on the importance of the location of their complex zeros.
A few of the most important names in the subsequent study of these number
theoretic functions are those of Dedekind, Hilbert, Hecke, Artin, Weil,
and Tate.
Ignorance prevents me from entering into a discussion of these
functions and what is known about them. However, it seems that they
provide some of the best reasons for believing that the Riemann hypothesis
is true  for believing, in other words, that there is a profound and
as yet uncomprehended numbertheoretic phenomenon, one facet of which is
that the roots all lie on Re[s] = 1/2. In particular, there is a 'zeta
function' associated in a natural numbertheoretic way to any function field over
a finite field, and Weil has shown that the analog of the
Riemann hypothesis is true for such 'zeta functions'"
H.M. Edwards, from Riemann's Zeta Function (Academic Press, 1974)
p. 298
Some useful discussion of the very
fundamental question "what is a zeta function?" took place at the
2002 conference on zeta functions and
associated Riemann Hypotheses in New York.
As M. Huxley puts it: "We know one when we see one."
J. Baez's This Weeks
Finds in Mathematical Physics Week 216 contains some useful discussion about various
types of zeta functions. week 217
includes very helpful discussion of the Riemann Hypothesis, Extended Riemann Hypothesis, Grand
Riemann Hypothesis, Weil Conjectures, Langlands Programme, the functional equations
of zeta and Lfunctions, modularity of theta functions, etc.
week 218
follows this up, framing certain issues concerning the bewildering array of zeta
and Lfunctions in terms of category theory.
P.E. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.), Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)
Probably the best overall view of zeta functions as a general phenomenon appears in
the Mathematical Society of Japan's Encyclopedic Dictionary of Mathematics (pp.
13721392 of the English edition published by MIT Press in 1977)
The introductory section informs us:
"Since the 19th century, many special functions called zeta functions have been
defined and investigated. The four main problems concerning zeta functions are:
(1) Creation of new zeta functions.
(2) Investigation of the properties of zeta functions.
Generally, zeta functions have the following four properties in common: (i) They
are meromorphic on the whole complex plane; (ii) they have Dirichlet series expansions;
(iii) they have Euler products expansions; and (iv) they satisfy certain functional
equations. Also it is an important problem to find the poles, residues, and zeros of
zeta functions.
(3) Application to number theory, in particular to the theory of
decomposition of prime ideals in finite extensions of algebraic number fields. (4)
Study of the relations between different zeta functions.
Most of the functions called zetafunctions or Lfunctions
have the four properties of problem (2). The following is a classification of the important types
of zeta functions that are already known, which will be discussed later in this article:
(1) The zeta and Lfunctions of algebraic number fields: the Riemann zeta
function, Dirichlet Lfunctions (study of these functions gave impetus to the
theory of zeta functions), Dedekind zeta functions, Hecke Lfunctions,
Hecke Lfunctions with Grossencharakters, Artin Lfunctions, and Weil
Lfunctions.
(2) The padic Lfunctions related to the works
of H.W. Leopoldt, T. Kubota, K. Iwasaw, etc.
(3) The zeta functions of quadratic forms:
Epstein zeta functions, zeta functions of indefinite quadratic forms (C.L. Siegel),
etc.
(4) The zeta and Lfunctions of simple algebras: Hey zeta functions and the zeta functions
given by R. Godement, T. Tamagawa, etc.
(5) The zeta functions associated with Hecke
operators, related to the work of E. Hecke, M. Eichler, G. Shimura, etc.
(6) The congruence zeta and Lfunctions
attached to algebraic varietites defined over finite fields (E. Artin, F.K. Schmidt, A. Weil);
Hasse zeta functions attached to the algebraic varietites defined over algebraic number
fields.
(7) The zeta functions attached to discontinuous groups: Selberg zeta functions,
the Eisenstein series defined by A. Selberg, Godement, and I.M. Gel'fand, etc.
(8) Y. Ihara's
zeta function related to nonAbelian class field theory over a function field over a
finite field.
(9) zeta functions associated with prehomogeneous vector spaces."
alphabetic directory
zeta function of an Abelian variety
G. Shimura,
Abelian Varieties with Complex Multiplication and
Modular Functions (Princeton, 1997)
zeta function of an algebraic curve over a finite field
M. Deurling, "The zetafunctions of algebraic curves and varieties", Report of
an International Colloquium on ZetaFunctions (1956), K. Chandrasekharan, editor.
W. A. ZúñigaGalindo,
"Zeta functions of singular curves over finite fields", Revista Colombiana de Matematicas 31 (1997)
115124.
zeta functions for Anosov flows
D. Ruelle, "Zeta functions for expanding maps and Anosov flows",
Inventiones Math. 34 (1976) 231242.
ArtinMazur zeta function
D. Ruelle, "Dynamical zeta functions and transfer operators"
Ruelle explains that ArtinMazur zeta functions are Weil zeta functions in the case where we have a diffeomorphism on a compact manifold.
D. Ruelle, "Zeta functions and statistical mechanics", Asterisque 40 (1976), 167176.
ArtinWeil zeta function
DFG research group  "Transfer operators and dynamical zeta functions"
zeta function of an attractor
R. Williams,
"The zeta function of an attractor", Conference on the Topology
of Manifolds, (editors J.C. Hocking, et. al.) (1968), 155161.
zeta functions of automorphisms of free groups
M. Lustig, et. al.,
Geodesics on flat surfaces with conical singularities/ zeta function
for automorphisms of free groups
Barnes zeta function
J.S. Dowker and K. Kirsten, "The Barnes zetafunction, sphere
determinants and Glaisher KinkelinBendersky constants"
Beurling modified zeta functions for systems of gprimes
E. Stankus, "Modified zeta functions and the
number of gintegers"
Beurling gprimes bibliography and
notes
Burgess zeta functions
"The main online reference (also mentioned in [2]) appears to be the 2009 paper
by Terry Tao: A remark on partial sums involving the Moebius function [3], in
which he defines them as analagous to the Riemann zeta function, but
acting only over the multiplicative semigroup generated by a given set of
primes $\mathcal{P}$: $\zeta_\mathcal{P}$ is then defined for $Re(s) > 1$ by the formula
$\zeta_\mathcal{P}(s) := \sum_{n \is <\mathcal{P}>}{\frac{1}{n^s}}
= \prod_{p \is \mathcal{P}}{ (1  \frac{1}{p^s})^{1}}$
[2] http://mathoverflow.net/questions/28000/whataretheanalyticpropertiesofdirichleteulerproductsrestrictedtoarithm
[3] http://arxiv1.library.cornell.edu/pdf/0908.4323
[Hugo van der Sanden]
Carlitz zeta function
J.P. Allouche,
"Finite automata and arithmetic"
G. Damamme and Y. Hellegouarch, "Transcendence of the values of the
Carlitz zeta function by Wade's method", Journal of Number Theory
39 (1991) 257278.
zeta functions for crystallographic groups
M. du Sautoy,
J. McDermott, and G. Smith, "Zeta functions of crystallographic groups
and analytic continuation", Proceedings of the London Mathematical
Society 79 (3) (1999) 511534.
zeta functions associated with curves over finite fields
Artin conjectured that the Riemann hypothesis holds for the zeta function
associated with any curve over a finite field. Hasse proved this for
elliptic curves. Andre Weil (while in prison during WWII) proved it for
arbitrary curves. Deligne went on to prove it for general algebraic
varieties.
zeta functions of cyclotomic fields
http://www.princeton.edu/~missouri/Generals/generals/prasanna_kartik
Dedekind zeta functions
WWN notes on Dedekind zeta functions
(part of a workinprogress)
From J.T. Tate, "Fourier analysis in number fields and Hecke's
zetafunctions" [1950 Princeton Ph.D. thesis, reproduced as Chapter 15 of Algebraic
Number Theory by J.W.S. Cassels and A. Fröhlich (Academic Press, 1967)]:
"Hecke was the first to prove that the Dedekind zetafunction of
any algebraic number field has an analytic continuation over the
whole plane and satisfies a simple functional equation. He soon realized
that this method would work, not only for the Dedekind zetafunction and
Lseries, but also for a zetafunction formed with a new type of ideal
character which, for principal ideals depends not only on the residue class
of the number modulo the "conductor", but also on the position of the
conjugates of the number in the complex field. Overcoming rather
extraordinary technical complications, he showed (1918 and 1920) that
these "Hecke" zetafunctions satisfied the same type of functional
equation as the Dedekind zetafunction, but with a much more complicated
factor."
P. Cohen,
Dedekind zeta functions and quantum statistical mechanics
[abstract] "In this article we construct a C*dynamical system with partition
function the Dedekind zeta function of a given number field and with
a phase transition at the pole of this zeta function which detects a
breaking of symmetry with respect to a natural symmetry group.
This extends work of BostConnes and HarariLeichtnam."
http://at.yorku.ca/cgibin/amca/cadx04
See also Appendix A.1 of Fractal Geometry and Number Theory
by M.L. Lapidus and M. van Frankenhuysen
zeta function of a discrete group of automorphisms of a bounded degree tree
http://euler.slu.edu/Dept/Faculty/clair/papers.html
zeta function of a distributive lattice
K.H. Knuth, "Lattice duality: The origin of probability
and entropy", Neurocomputing 67 C (2005) 245274
zeta function of a division algebra
T. Tamagawa, "On the zeta function of a division algebra",
Annals of Mathematics 77 (1963) 387405.
dynamical zeta functions
number theory and physics archive page
A. Juhl,
Cohomological Theory of Dynamical Zeta Functions (Progress in Mathematics,
Vol. 194.) (Birkhauser, 2001)
R. Mainieri, Arithmetical properties of dynamical zeta
functions
http://www.math.psu.edu/dynsys/semSpring00.html
http://www.geom.umn.edu/~rminer/talks/cecm/ttmath/Ruelle2.html
http://www.ma.man.ac.uk/~mp/research.html
Elizalde zeta functions
A class of zeta functions that extends the class of Epstein's was
recently brought to my attention by
Prof. E. Elizalde of M.I.T. Although I don't think they've appeared
in print under this name, it seems an appropriate one to give them. They
are spectral zeta functions associated with a quadratic + linear + constant
form in any number of dimensions. Elizalde has developed formulas
for them which extend the famous ChowlaSelberg formula.
E. Elizalde, "Explicit zeta functions for bosonic and fermionic
fields on a noncommutative toroidal spacetime", Journal of Physics
A 34 (2001) 30253036.
E. Elizalde, "Multidimensional extension of the generalized
ChowlaSelberg formula", Communications in Mathematical Physics
198 91998) 8395.
E. Elizalde, "Zeta functions, formulas and applications",
J. Comp. Appl. Math. 118 (2000) 125.
zeta functions of elliptic operators
S. Moronianu, "Adiabatic
limits of eta and zeta functions of elliptic operators"
http://www.math.ohiostate.edu/Graduate/THESIS_ABSTRACTS/BUCICOVSCHI.BOGDAN.html
zeta functions of energy of PTsymmetric quantum systems
http://www.physics.wustl.edu/graduate/archive/Wang111000.html
Epstein zeta function
Heilbronn proved that the Riemann Hypothesis fails for the Epstein
zeta function.
Eric Weisstein's
notes
S. Chowla and A. Selberg, "On Epstein's zetafunction", J. Reine
und angew. math. 227 (1967) 86110.
D. Hejhal, "Zeros of Epstein zetafunctions and supercomputers",
Proc. Intern. Congress. Math. (Berkeley, 1986) Vol. II (AMS,
1987) 13621384.
C. Siegel, "A generalization of the Epstein zeta function", Report of
an International Colloquium on ZetaFunctions (1956), K. Chandrasekharan, editor.
U. Christian, Selberg's Zeta, L and Eisensteinseries (Lecture
Notes in Mathematics 1030, Springer, 1983)
http://www.aurora.edu/~ldelacey/vita2.htm
Appendix A.4 of Fractal Geometry and Number Theory
by M.L. Lapidus and M. van Frankenhuysen
M.L. Glasser and I.J. Zucker, "Lattice Sums in Theoretical Chemistry." In Theoretical
Chemistry: Advances and Perspectives, Vol. 5 (Ed. H. Eyring). New York: Academic Press,
pp. 6970, 1980.
D. Shanks, "Calculation and Applications of Epstein Zeta Functions." Math. Comput.
29 (1975) 271287.
Estermann zeta function
http://www.fsci.fuk.kindai.ac.jp/~kanemitu/number.html
http://www.mscs.dal.ca/~dilcher/berni.html
Euler zeta function
as defined in the Prime Pages glossary
K. Devlin, "How Euler discovered the zeta function"
(elementary historical introduction)
zeta functions for expanding maps
D. Ruelle, "Zeta functions for expanding maps and Anosov flows",
Inventiones Math. 34 (1976) 231242.
zeta function associated with finite extension of the rational numbers
http://www.cs.bgu.ac.il/~saarh/colloquium/goren/goren.html
zeta functions for flows
D. Ruelle, "Zeta functions and statistical
mechanics", Asterisque 40 (1976), 167176.
A. Juhl,
Cohomological Theory of Dynamical Zeta Functions (Progress in Mathematics,
Vol. 194.) (Birkhauser, 2001)
zeta functions for forms of Fermat equations
L. Brünjes, Forms of
Fermat Equations and their Zeta Functions (World Scientific, 2004)
zeta function of a generalised cone
http://wwwphys.science.unitn.it/research/consuntivi/ft97campi.html
geometric zeta functions
http://www.best.com/~worktree/g/87/243g.htm
publications of M.L. Lapidus
A. Deitmar, "Geometric zetafunctions on padic groups"
A. Deitmar, "Geometric zetafunctions of locally symmetric spaces", Am. J. Math. 122 vol.5 (2000) 887926.
A. Deitmar, "Geometric zetafunctions, L^{2}theory,
and compact Shimura manifolds"
Goss zeta function
http://www.math.uiuc.edu/AlgebraicNumberTheory/0096/
zeta function of a finite graph
A. Terras, Zeta Functions of Graphs: A Stroll Through the Garden (Cambridge Univ. Press, 2010)
A. Terras and H. Stark, "Zeta functions of finite graphs and coverings", Advances in Mathematics
121 (1996) 124165.
A. Terras and H. Stark, "Zeta functions of finite graphs and coverings, Part II", Advances in Mathematics,
154 (2000), 132195.
http://www.math.dartmouth.edu/~colloq/s97/stark.html
zeta function of a finite unoriented graph
see IharaSelberg zeta function
GutzwillerVoros zeta function
http://www.mpg.de/reports/9814/9814_T.htm
http://130.83.24.4/nhc/activities/ChaoticScattering.html
HasseWeil zeta function (of an elliptic curve)
http://www.dpmms.cam.ac.uk/AlgebraicNumberTheory/0095/
http://www.math.purdue.edu/research/seminars/old_abstracts/1999/ html/abs_11_30_99a.html
Hawking zeta function
http://www.oa.uj.edu.pl/~maslanka/
http://citeseer.nj.nec.com/114922.html
http://citeseer.nj.nec.com/smith95fundamental.html
Hecke zeta functions
"Hecke was the first to prove that the Dedekind zetafunction of
any algebraic number field has an analytic continuation over the
whole plane and satisfies a simple functional equation. He soon realized
that this method would work, not only for the Dedekind zetafunction and
Lseries, but also for a zetafunction formed with a new type of ideal
character which, for principal ideals depends not only on the residue class
of the number modulo the "conductor", but also on the position of the
conjugates of the number in the complex field. Overcoming rather
extraordinary technical complications, he showed (1918 and 1920) that
these "Hecke" zetafunctions satisfied the same type of functional
equation as the Dedekind zetafunction, but with a much more complicated
factor."
From J.T. Tate, "Fourier analysis in number fields and Hecke's
zetafunctions" [1950 Princeton Ph.D. thesis, reproduced as Chapter 15 of Algebraic
Number Theory by J.W.S. Cassels and A. Fröhlich (Academic Press, 1967)]
height zeta functions
J. Shalika and Y. Tschinkel, "Height
zeta functions of equivariant compactifications of the Heisenberg group"
Hey zeta functions
P. Roquette, "Class field theory in
characteristic p, its origin and development"
Hlawka zeta function
E. Hlawka, "Uber die Zetafunktion konvexer Körper", Monatsh.
Math. 54 (1950) 8199.
The term "Hlawka's zetafunction" has recently used by (among
others) Martin Huxley.
zeta function of a homeomorphism
http://www.math.nwu.edu/graduate/prelims/dyna88.pdf
Hurwitz zeta function
J. Borwein, D. Bradley and R. Crandall, "Computational strategies
for the Riemann zeta function", J. Comp. App. Math. 121 (2000) p.8
defined as a generalisation of Riemann's zeta function
Eric Weisstein's
notes
V. Adamchik,
"Derivatives of the Hurwitz zeta function for rational arguments",
Journal of Computational and Applied Mathematics 100
(1999) 201206.
A. Veselov and J. Ward,
"On the real roots of the Bernoulli polynomials and the Hurwitz zetafunction"
(1999)
J. Andersson,
"Mean value properties of the Hurwitz zetafunction",
Mathematica Scandinavica 71 (1992) 295300.
V. Adesi and S. Zerbini, "Anayltic
continuation of the Hurwitz zeta function with physical applications"
O. Espinosa and V. Moll, "On
some integrals involving the Hurwitz zeta function: part 2"
M. Katsurada and K. Matsumuto, "Explicit formulas and asymptotic
expansions for certain mean square of Hurwitz zetafunctions",
Proc. Japan Acad. 69 (8) (1993) 303307.
V.V. Rane, "On Hurwitz zetafunction", Math. Ann. 264
(2) 147151.
W.P. Zhang, "On the mean square value of the Hurwitz zetafunction",
Illinois Journal of Mathematics 38 (1) (1994) 7178.
Igusa local zeta function
http://www.mtholyoke.edu/~robinson/reu/reu95/reu95.html
POLYGUSA  Computer program to
calculate Igusa's local zeta function associated to a polynomial
IharaHashimotoBass zeta function
http://euler.slu.edu/Dept/Faculty/clair/clair.html
IharaSelberg zeta function (zeta function of a finite unoriented graph)
D. Ruelle, "Dynamical zeta functions and transfer operators", p.4
Ruelle explains that they are defined in terms of the Euler product formula for Weil zeta functions, where periodic orbits are replaced by cycles (circuits on the graph in question without immediate backtracking). The reciprocals of these zeta functions are known to be polynomials, and the functions themselves are known to satisfy Riemann hypotheses precisely when the graph in question is Ramanujan.
http://www.pdmi.ras.ru/preprint/2000/0007.html
H. Bass, "The IharaSelberg zeta function of a tree lattice", Int. J. Math.3 No. 6 (1992) 717797.
Incomplete (Riemann) zeta function
K.S. Kolbig,"Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function",
Math. Comput. 24 (1970) 679696.
[Jacobi zeta function]
M. Somos points out "...not at all like the
other zeta functions mentioned. It is an elliptic function
with double quasiperiodicity. There is nothing like the
nontrivial zeros a la Riemann Hypothesis...note that it is just a historical accident that it was
called a zeta function and has nothing to do with the rest...As a curiousity, the Jacobi theta function is involved
with the functional equation of the Riemann zeta function
via the Mellin transform. However, this is as close as it
gets regarding Jacobi and his elliptic functions."
Köhler zeta functions
http://www.mathematik.unileipzig.de/GK/GKKolloquium.html
R. Berndt, "Köhler's computation of his Zeta function for an arithmetic curve of degree two",
Mitt. Math. Ges. Hamburg III (Hamburg, 1985)
zeta functions associated with Laplacetype operators
http://www.na.infn.it/gravity2001/qgprogram.htm
Kurokawa multiple zeta functions
N. Kurokawa, "Multiple zeta functions: an example", Adv.
Studies in Pure Math., Zeta functions in geometry (1991)
Lefschetz zeta function
D. Ruelle, "Dynamical zeta functions and transfer operators"
Ruelle defines the Lefschetz zeta function analogously to the Weil zeta function, except fixed points are weighted by their topological indices. He points out that in many interesting cases all topological indices equal 1, in which case the Lefschetz zeta function becomes identical to the Weil zeta function.
Lerch zeta function
http://www.mif.vu.lt/~garunkstis
J. Borwein, D. Bradley and R. Crandall, "Computational strategies
for the Riemann zeta function", J. Comp. App. Math. 121 (2000) p.11
R. Garunkstis and A. Laurincikas, "The Lerch zetafunction", Integral Transforms
and Special Functions 10 (34) (2000) 211226
J. Ignataviciute, "A limit theorem for the Lerch zetafunction", Special issue of Lietuvos Matematikos Rinkinys 40 (2000):
Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai,
June 2223, 2000, 2127
A. Laurincikas, "On the mean square of the Lerch zetafunction with respect to the
parameter", Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai, June 2223, 2000, 4348
A. Laurincikas and K. Matsumoto, "The joint universality and the functional
independence for Lerch zetafunctions", Nagoya Journal of Mathematics 157
(2000) 211227
zeta function of Lyapunov exponent of a product of random matrices
http://ups.cs.odu.edu/buckets/ups.xxx.chaodyn/xxx.xxx.chaodyn.9301001/
Matsumoto zeta function
R. Kacinskaite, "A discrete limit theorem for the Matsumoto zetafunction on
the complex plane", Lietuvos Matematikos Rinkinys
40 (4) (2000) 475492, (in Russian) Lithuanian Mathematical Journal
40(4) (2000) 364378.
R. Kacinskaite, "On the value distribution of Matsumoto zetafunction on the
complex plane", Special issue of Lietuvos Matematikos Rinkinys 40 (2000):
Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai,
June 2223, 2000, 3338.
MinakshisundaramPleijel zeta function
http://mmf.ruc.dk/~Booss/recoll.pdf
S. Minakshisundaram and A. Pleijel, "Some properties of the
eigenfunctions of the Lapalace operator on Riemannian manifolds",
Canadian Journal of Mathematics 1 (1949) 242256.
H.P. McKean, "Selberg's trace formula as applied to a compact
Riemann surface", Communications on Pure and Applied Mathematics
25 (1972) 225246.
motivic zeta function
http://www.wis.kuleuven.ac.be/wis/algebra/NotesCambridge/Naive%20motivic%20zeta%20function.htm
http://cwisdb.cc.kuleuven.ac.be/research/P/3E98/project3E980397.htm
multiplesum zeta functions
E. Elizalde, "Multiple zeta functions with arbitrary exponents", Journal of Physics
A 22 (1989) 931942.
Nielsen zeta function
http://www.yurinsha.com/317/ws11.1.htm
zeta function associated with nilpotent group
http://muse.jhu.edu/demo/ajm/
Nint zeta function
Eric Weisstein's
notes
J.M. Borwein, et.al., "Nearest Integer ZetaFunctions"
Amer. Math. Monthly 101 (1994) 579580.
NonAbelian zeta functions
L. Weng, "Constructions of NonAbelian Zeta Functions for Curves"
L. Weng, "Refined BrillNoether Locus and NonAbelian Zeta Functions for Elliptic Curves"
L. Weng, "RiemannRoch, Stability and New NonAbelian Zeta Functions for Number Fields"
L. Weng, " New NonAbelian Zeta Functions for Curves over Finite Fields"
padic zeta function
http://wwwfourier.ujfgrenoble.fr/AIF/Vol38/E383_1/E383_1.html
partial zeta functions
D. Wan, "Partial
zeta functions of algebraic varieties over finite fields"
J.P. Jurzak, "Partial
Euler products as a new approach to Riemann Hypothesis"
periodic zeta function
Eric Weisstein's
notes
A. Kacenas and A. Laurincikas, "A note on the valuedistribution of the periodic
zetafunction", Special issue of Lietuvos Matematikos Rinkinys 40 (2000):
Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai,
June 2223, 2000, 2832.
zeta function of Picard modular surfaces
http://www.math.ias.edu/~goresky/publ.html
zeta functions for piecewise monotonic transformations
http://www.math.chs.nihonu.ac.jp/~mori/lectures.html
zeta functions related to polyBernoulli numbers
http://www.math.kindai.ac.jp/math/ohno/ohnore.html
zeta function of certain prehomogeneous vector spaces
http://ups.cs.odu.edu/buckets/ups.xxx.math/xxx.xxx.math.9408212/
prime zeta function
Eric Weisstein's
notes
probablilistic generalisation of the Riemann zeta function
N. Boston,
"A probabilistic generalization of the Riemann zeta function",
Analytic Number Theory, Vol. 1, Progr. Math. 138,
(Birkhauser, 1996) 155162.
qanalogues of the Riemann zeta function
I. Cherednik,
"On
qanalogues of Riemann's zeta"
M. Kaneko, N. Kurokawa, and M. Wakayama,
"A variation of Euler's
approach to values of the Riemann zeta function"
[abstract:] "An elementary method of computing the values at negative integers of the
Riemann zeta function is presented. The principal ingredient is a new qanalogue of the
Riemann zeta function. We show that for any argument other than 1 the classical limit of
this qanalogue exists and equals the value of the Riemann zeta."
Redei zeta function
J.P.S. Kung, M. Ram Murty, G.C. Rota, "On the Redei zeta function", J. Number Theory 12 (1980) 421436
zeta function of a regular language
http://theory.lcs.mit.edu/~dmjones/hbp/tcs/Authors/honkalajuha.html
Reidemeister zeta function
http://www.yurinsha.com/317/ws11.1.htm
Riemann zeta function ("the grandmother of all zeta functions"  D. Ruelle)
number theory and physics archive page
Ruelle zeta function
S.J. Patterson, "On Ruelle's zeta function", Israel Math. Conf.
Proc. 3 (1990) 163184.
A. Juhl,
Cohomological Theory of Dynamical Zeta Functions (Progress in Mathematics,
Vol. 194.) (Birkhauser, 2001)
dynamical and spectral zeta functions archive page
http://www.nbi.dk/CATS/c_e_borel/steiner_course
http://www.geom.umn.edu/~rminer/talks/cecm/ttmath/Ruelle2.html
Selberg zeta function
number theory and physics archive page
Eric Weisstein's
notes
A. Voros, "Spectral functions and the Selberg zeta function",
Communications in Mathematical Physics 110 (1987)
439465.
A. Juhl,
Cohomological Theory of Dynamical Zeta Functions (Progress in Mathematics,
Vol. 194.) (Birkhauser, 2001)
U. Christian, Selberg's Zeta, L and Eisensteinseries (Lecture
Notes in Mathematics 1030, Springer, 1983)
In this book, the author proves the analytic continuation and functional
equation for the Selberg zeta function.
Iwaniec, H., Introduction to the Spectral Theory of Automorphic Forms,
2^{nd} edition, Graduate Studies in Mathematics 53 (AMS, 2002)
Chapter 10 covers the Trace Formula, and on p.154 we find a helpful note about
Selberg zeta functions:
"If you will, the Selberg zetafunction satisfies an analogue of the Riemann
hypothesis. However, the analogy with the Riemann zetafunction
is superficial. First of all, the Selberg zeta function has no natural development into
Dirichlet series. Furthermore, the functional equation...resists any decent interpretation as a kind of
Poisson summation principle. Nevertheless, modern studies of Z(s) have caused
a lot of excitement in mathematical physics (see [Sa1]). At least, one may say that the dream
of Hilbert and Pólya of connecting the zeros of a zetafunction with eigenvalues of a selfadjoint
operator is a reality in the context of Z(s)."
[Sa1] P. Sarnak, "Determinants of Laplacians", Communications in Mathematical Physics
110 (1987) 113120.
http://www.wileyvch.de/books/tis/eng/3527400729.html
http://www.nbi.dk/CATS/c_e_borel/steiner_course
http://www.isibang.ac.in/Smubang/as/publi.htm
semisimple zeta function of quaternionic Shimura varieties
http://206.67.72.201/catalog/np/may97np/DATA/354062645x.html
Shintani zeta function
A. Yukie, Shintani Zeta Functions (LMS Lecture Note Series 183,
Cambridge University Press, 1993)
"The purpose of this book is to introduce an approach based on geometric
invariant theory to the global theory of zeta functions for prehomogeneous
vector spaces."
http://www.math.okstate.edu/preprint/1995.html
zeta function of a simplicial complex
http://citeseer.nj.nec.com/orner96subspace.html
zeta function of singular curve over a finite field
http://www.emis.de/journals/RCM/vol312/97310206.html
Solomon's zeta function
http://www.wits.ac.za/science/number_theory/jplkpub.htm
spectral zeta function
http://www.maths.ex.ac.uk/~mwatkins/zeta/physics3.htm
http://journals.wspc.com.sg/mpla/preserveddocs/132/gon.pdf
stochastic zeta function (of a shift)
http://www.math.washington.edu/~lind/Papers/spantree.pdf
zeta function of a stochastic matrix
notes from James Propp
thermodynamic zeta functions
R. Mainieri, Arithmetical properties of dynamical zeta functions
M. Holthaus and E. Kalinowski,
"Condensate fluctations in trapped Bose gases: Canonical vs. microcanonical ensemble", Annals of Physics 270 (1998) 198230.
M. Holthaus, K.T. Kapale, V.V. Kocharovsky and M.O. Scully,
"Master equation vs. partition function: canonical statistics of ideal BoseEinstein condensates", Physica A 300 (2001) 433467.
topological zeta functions
http://www.wis.kuleuven.ac.be/wis/algebra/NotesCambridge/Topological%20zeta%20function.htm
W. Veys, "Determination of the poles of the topological zeta function for curves",
Manuscripta Math. 87 (1995), 435448
W. Veys, "The topological zeta function associated to a function on a normal surface germ",
Topology 38 (1999) 439456
D. Segers and W. Veys, "On the smallest poles of
topological zeta functions", Compositio Math. 140 (2004) 130144
D. Segers, "Smallest poles of Igusa's
and topological zeta functions and solutions of polynomial congruences"
A. Lemahieu, D. Segers and W. Veys, "On the poles of topological zeta functions",
preprint (2004), 11pp.
zeta functions of Turing machines
C.S. Calude and M. Stay, "Natural
halting probabilities, partial randomness, and zeta functions" (preprint 01/06)
twovariable zeta function for number fields
J.C. Lagarias,
"On a twovariable
zeta function for number fields"
van der GeerSchoof zeta function
http://at.yorku.ca/cgibin/amca/cadx67
zeta functions of varieties
M. Deurling, "The zetafunctions of algebraic curves and varieties", Report of
an International Colloquium on ZetaFunctions (1956), K. Chandrasekharan, editor.
B. Dwork, "On the rationality of the zeta function of an algebraic variety",
American Journal of Mathematics 82 (1960) 632648.
See also Weil zeta functions
http://www.math.berkeley.edu/~ribet/Colloquium/dwan.html
G. Shimura,
Abelian Varieties with Complex Multiplication and
Modular Functions (Princeton, 1997)
[Weierstrass zeta function]
M. Somos points out "...not at all like the
other zeta functions mentioned...just a variant of the Jacobi zeta function."
Weil zeta functions
D. Ruelle, "Dynamical zeta functions and transfer operators"
Ruelle defines the Weil zeta function for an algebraic variety over a finite field in terms of the numbers of fixed points of all iterations of the Frobenius map on the extension of the algebraic variety to the algebraic closure of the finite field. He goes on to explain how the concept can be extended to more general maps on more general spaces. Weil zeta functions have Euler product formulas over the set of periodic orbits.
D. Ruelle, "Zeta functions and statistical mechanics", Asterisque 40 (1976), 167176.
Witten zeta function
A. Reznikov, "Characteristic classes in symplectic topology",
Selecta Math. vol 3 (1997) 601642
http://xxx.lpthe.jussieu.fr/abs/math/9903178
http://www.math.columbia.edu/~gunnells/pubs/dedekind/dedekind.html
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