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 Chowla-Selberg formula.
E. Elizalde, "Explicit zeta functions for bosonic and fermionic
fields on a noncommutative toroidal spacetime", Journal of Physics
A 34 (2001) 3025-3036.
E. Elizalde, "Multidimensional extension of the generalized
Chowla-Selberg formula", Communications in Mathematical Physics
198 91998) 83-95.
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.ohio-state.edu/Graduate/THESIS_ABSTRACTS/BUCICOVSCHI.BOGDAN.html
zeta functions of energy of PT-symmetric 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 zeta-function", J. Reine
und angew. math. 227 (1967) 86-110.
D. Hejhal, "Zeros of Epstein zeta-functions and supercomputers",
Proc. Intern. Congress. Math. (Berkeley, 1986) Vol. II (AMS,
1987) 1362-1384.
C. Siegel, "A generalization of the Epstein zeta function", Report of
an International Colloquium on Zeta-Functions (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. 69-70, 1980.
D. Shanks, "Calculation and Applications of Epstein Zeta Functions." Math. Comput.
29 (1975) 271-287.
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) 231-242.
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), 167-176.
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://www-phys.science.unitn.it/research/consuntivi/ft97-campi.html
geometric zeta functions
http://www.best.com/~worktree/g/87/243g.htm
publications of M.L. Lapidus
A. Deitmar, "Geometric zeta-functions on p-adic groups"
A. Deitmar, "Geometric zeta-functions of locally symmetric spaces", Am. J. Math. 122 vol.5 (2000) 887-926.
A. Deitmar, "Geometric zeta-functions, L2-theory,
and compact Shimura manifolds"
Goss zeta function
http://www.math.uiuc.edu/Algebraic-Number-Theory/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) 124-165.
A. Terras and H. Stark, "Zeta functions of finite graphs and coverings, Part II", Advances in Mathematics,
154 (2000), 132-195.
http://www.math.dartmouth.edu/~colloq/s97/stark.html
zeta function of a finite unoriented graph
see Ihara-Selberg zeta function
Gutzwiller-Voros zeta function
http://www.mpg.de/reports/9814/9814_T.htm
http://130.83.24.4/nhc/activities/ChaoticScattering.html
Hasse-Weil zeta function (of an elliptic curve)
http://www.dpmms.cam.ac.uk/Algebraic-Number-Theory/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 zeta-function 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 zeta-function and
L-series, but also for a zeta-function 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" zeta-functions satisfied the same type of functional
equation as the Dedekind zeta-function, but with a much more complicated
factor."
From J.T. Tate, "Fourier analysis in number fields and Hecke's
zeta-functions" [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) 81-99.
The term "Hlawka's zeta-function" 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) 201-206.
A. Veselov and J. Ward,
"On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function"
(1999)
J. Andersson,
"Mean value properties of the Hurwitz zeta-function",
Mathematica Scandinavica 71 (1992) 295-300.
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 zeta-functions",
Proc. Japan Acad. 69 (8) (1993) 303-307.
V.V. Rane, "On Hurwitz zeta-function", Math. Ann. 264
(2) 147-151.
W.P. Zhang, "On the mean square value of the Hurwitz zeta-function",
Illinois Journal of Mathematics 38 (1) (1994) 71-78.
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
Ihara-Hashimoto-Bass zeta function
http://euler.slu.edu/Dept/Faculty/clair/clair.html
Ihara-Selberg 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/2000/00-07.html
H. Bass, "The Ihara-Selberg zeta function of a tree lattice", Int. J. Math.3 No. 6 (1992) 717-797.
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) 679-696.
[Jacobi zeta function]
M. Somos points out "...not at all like the
other zeta functions mentioned. It is an elliptic function
with double quasi-periodicity. There is nothing like the
non-trivial 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.uni-leipzig.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 Laplace-type 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 zeta-function", Integral Transforms
and Special Functions 10 (3-4) (2000) 211-226
J. Ignataviciute, "A limit theorem for the Lerch zeta-function", Special issue of Lietuvos Matematikos Rinkinys 40 (2000):
Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai,
June 22-23, 2000, 21-27
A. Laurincikas, "On the mean square of the Lerch zeta-function with respect to the
parameter", Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai, June 22-23, 2000, 43-48
A. Laurincikas and K. Matsumoto, "The joint universality and the functional
independence for Lerch zeta-functions", Nagoya Journal of Mathematics 157
(2000) 211-227
zeta function of Lyapunov exponent of a product of random matrices
http://ups.cs.odu.edu/buckets/ups.xxx.chao-dyn/xxx.xxx.chao-dyn.9301001/
Matsumoto zeta function
R. Kacinskaite, "A discrete limit theorem for the Matsumoto zeta-function on
the complex plane", Lietuvos Matematikos Rinkinys
40 (4) (2000) 475-492, (in Russian) Lithuanian Mathematical Journal
40(4) (2000) 364-378.
R. Kacinskaite, "On the value distribution of Matsumoto zeta-function on the
complex plane", Special issue of Lietuvos Matematikos Rinkinys 40 (2000):
Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai,
June 22-23, 2000, 33-38.
Minakshisundaram-Pleijel 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) 242-256.
H.P. McKean, "Selberg's trace formula as applied to a compact
Riemann surface", Communications on Pure and Applied Mathematics
25 (1972) 225-246.
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
multiple-sum zeta functions
E. Elizalde, "Multiple zeta functions with arbitrary exponents", Journal of Physics
A 22 (1989) 931-942.
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 Zeta-Functions"
Amer. Math. Monthly 101 (1994) 579-580.
Non-Abelian zeta functions
L. Weng, "Constructions of Non-Abelian Zeta Functions for Curves"
L. Weng, "Refined Brill-Noether Locus and Non-Abelian Zeta Functions for Elliptic Curves"
L. Weng, "Riemann-Roch, Stability and New Non-Abelian Zeta Functions for Number Fields"
L. Weng, " New Non-Abelian Zeta Functions for Curves over Finite Fields"
p-adic zeta function
http://www-fourier.ujf-grenoble.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 value-distribution of the periodic
zeta-function", Special issue of Lietuvos Matematikos Rinkinys 40 (2000):
Proceedings of XLI Conference of Lithuanian Mathematical Society, Ŝiauliai,
June 22-23, 2000, 28-32.
zeta function of Picard modular surfaces
http://www.math.ias.edu/~goresky/publ.html
zeta functions for piecewise monotonic transformations
http://www.math.chs.nihon-u.ac.jp/~mori/lectures.html
zeta functions related to poly-Bernoulli 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) 155-162.
q-analogues of the Riemann zeta function
I. Cherednik,
"On
q-analogues 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 q-analogue of the
Riemann zeta function. We show that for any argument other than 1 the classical limit of
this q-analogue 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) 421-436
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) 163-184.
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)
439-465.
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,
2nd 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 zeta-function satisfies an analogue of the Riemann
hypothesis. However, the analogy with the Riemann zeta-function
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 zeta-function with eigenvalues of a self-adjoint
operator is a reality in the context of Z(s)."
[Sa1] P. Sarnak, "Determinants of Laplacians", Communications in Mathematical Physics
110 (1987) 113-120.
http://www.wiley-vch.de/books/tis/eng/3-527-40072-9.html
http://www.nbi.dk/CATS/c_e_borel/steiner_course
http://www.isibang.ac.in/Smubang/as/publi.htm
semi-simple zeta function of quaternionic Shimura varieties
http://206.67.72.201/catalog/np/may97np/DATA/3-540-62645-x.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/vol31-2/97310206.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/preserved-docs/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) 198-230.
M. Holthaus, K.T. Kapale, V.V. Kocharovsky and M.O. Scully,
"Master equation vs. partition function: canonical statistics of ideal Bose-Einstein condensates", Physica A 300 (2001) 433-467.
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), 435-448
W. Veys, "The topological zeta function associated to a function on a normal surface germ",
Topology 38 (1999) 439-456
D. Segers and W. Veys, "On the smallest poles of
topological zeta functions", Compositio Math. 140 (2004) 130-144
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)
two-variable zeta function for number fields
J.C. Lagarias,
"On a two-variable
zeta function for number fields"
van der Geer-Schoof zeta function
http://at.yorku.ca/cgi-bin/amca/cadx-67
zeta functions of varieties
M. Deurling, "The zeta-functions of algebraic curves and varieties", Report of
an International Colloquium on Zeta-Functions (1956), K. Chandrasekharan, editor.
B. Dwork, "On the rationality of the zeta function of an algebraic variety",
American Journal of Mathematics 82 (1960) 632-648.
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), 167-176.
Witten zeta function
A. Reznikov, "Characteristic classes in symplectic topology",
Selecta Math. vol 3 (1997) 601-642
http://xxx.lpthe.jussieu.fr/abs/math/9903178
http://www.math.columbia.edu/~gunnells/pubs/dedekind/dedekind.html
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