We define two types of Witten’s zeta functions according to Cartan’s classification of compact symmetric spaces. The type II is the original Witten zeta function constructed by means of irreducible representations of the simple compact Lie group U. The type I Witten zeta functions, we introduce here, are related to the irreducible spherical representations of U. They arise in the harmonic analysis on compact symmetric spaces of the form U/K, where K is the maximal subgroup of U. To construct the type I zeta function we calculate the partition functions of 2d YM theory with broken gauge symmetry using the Migdal–Witten approach. We prove that for the rank one symmetric spaces the generating series for the values of the type I functions with integer arguments can be defined in terms of the generating series of the Riemann zeta-function.

1.
Cahn
,
R. S.
and
Wolf
,
J. A.
, “
Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one
,”
Comment. Math. Helvetici
51
,
1
21
(
1976
).
2.
Camporesi
,
R.
, “
On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact symmetric spaces of rank one
,”
J. Math. Anal. Appl.
214
(
2
),
524
549
(
1997
).
3.
Fine
,
D. S.
, “
Quantum Yang-Mills on a Riemann surface
,”
Commun. Math. Phys.
140
(
2
),
321
338
(
1991
).
4.
Gindikin
,
S.
and
Karpelevich
,
F. I.
, “
Plancherel measure for symmetric Riemannian spaces of non-positive curvature
,”
Dokl. Akad. Nauk SSSR
145
,
252
(
1962
).
5.
Harish-Chandra
, “
Spherical functions on a semisimple Lie group, I
,”
Am. J. Math.
80
,
241
310
(
1958
).
6.
Helgason
,
S.
,
Differential Geometry, Lie Groups, and Symmetric Spaces
(
Academic Press
,
1979
).
7.
Helgason
,
S.
, “
Harish-Chandra c-function, a Mathematical Jewel, A.
,” in
The mathematical Legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis: An AMS special Session honoring the memory of Harish-Chandra, January 9-10, 1998, Baltimore, Maryland
(
American Mathematical Society, Providence, RI
,
2000
), Vol.
68
, p.
273
.
8.
Ikeda
,
A.
, “
Spectral zeta functions for compact symmetric spaces of rank one
,”
Kodai Math. J.
23
(
3
),
345
357
(
2000
).
9.
Matsumoto
,
K.
, “
On Mordell-Tornheim and other multiple zeta-functions
,” in
Proceedings of the Session in Analytic Number Theory And Diophantine Equations
(
Bonner Mathematische Schriften
,
2003
), Vol.
360
;
Matsumoto
,
K.
and
Tsumura
,
H.
, “
On Witten multiple zeta-functions associated with semisimple Lie algebras I
,”
Ann. Inst. Fourier
56
(
5
),
1457
(
2006
);
Komori
,
Y.
,
Matsumoto
,
K.
, and
Tsumura
,
H.
, “
On Witten multiple zeta-functions associated with semisimple Lie algebras II
,”
J. Math. Soc. Jpn.
62
(
2
),
355
394
(
2010
);
Komori
,
Y.
,
Matsumoto
,
K.
, and
Tsumura
,
H.
, “
On Witten multiple zeta-functions associated with semisimple Lie algebras III
,”
Multiple Dirichlet Series, L-Functions and Automorphic Forms
(
Birkhäuser
,
Boston, MA
,
2012
), pp.
223
286
;
Komori
,
Y.
,
Matsumoto
,
K.
, and
Tsumura
,
H.
, “
On witten multiple zeta-functions associated with semi-simple lie algebras V
,”
Glasg. Math. J.
57
(
1
),
107
130
(
2015
);
Komori
,
Y.
,
Matsumoto
,
K.
, and
Tsumura
,
H.
, “
An introduction to the theory of zeta-functions of root systems
,” in
Algebraic and Analytic Aspects of Zeta Functions and L-Functions
,
MSJ Memoirs
21
, edited by
Bhowmik
,
G.
,
Matsumoto
,
K.
, and
Tsumura
,
H.
(
Mathematical Society of Japan
,
2010
), pp.
115
140
.
10.
Migdal
,
A. A.
, “
Recursion equations in gauge field theories
,”
30 Years of the Landau Institute - Selected Papers
(
World Scientific
,
1996
), pp.
114
119
.
11.
Minakshisundaram
,
S.
and
Pleijel
,
Å.
, “
Some properties of the Eigenfunctions of the Laplace-operator on Riemannian manifolds
,”
Can. J. Math.
1
(
3
),
242
256
(
1949
).
12.
Rusakov
,
B. Y.
, “
Loop averages and partition functions in U (N) gauge theory on two-dimensional manifolds
,”
Mod. Phys. Lett. A
05
(
09
),
693
703
(
1990
).
13.
Teo
,
L.-P.
, “
Zeta functions of spheres and real projective spaces
,” arXiv:1412.0758 (
2014
).
14.
Vilenkin
,
N. I.
,
Special Functions and the Theory of Group Representations
(
American Mathematical Society
,
1978
), Vol.
22
.
15.
Vretare
,
L.
, “
Elementary spherical functions on symmetric spaces
,”
Math. Scand.
39
(
2
),
343
358
(
1976
).
16.
Warner
,
G.
,
Harmonic Analysis on Semi-simple Lie Groups I
(
Springer Science and Business Media
,
2012
), Vol.
188
.
17.
Witten
,
E.
, “
On quantum gauge theories in two dimensions
,”
Commun. Math. Phys.
141
(
1
),
153
209
(
1991
).
18.
Zagier
,
D.
, “
Values of zeta functions and their applications
,”
First European Congress of Mathematics Paris
(
Birkhäuser Basel
,
1994
).
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