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Evaluations of higher depth determinants of Laplacians.

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Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
MSC 11M36
Evaluations of higher depth determinants of
Laplacians 1
c
Y. Yamasaki
Ehime University, Matsuyama, Japan
In this article, we evaluate higher depth determinants of Laplacians on the compact
Riemann surfaces with negative constant curvature. This is a summary of the results in
the forthcoming paper [5]
Keywords:
Hurwitz's zeta function, Selberg's zeta function, multiple gamma function,
polylogarithm function, determinants of Laplacians
џ 1. Introduction
Let T be an operator on some space. We assume that T has only discrete
spectrum ?0 6 ?1 6 . . . 6 ?n 6 . . . ? +? and the multiplicity of each eigenvalue
?j is nite. Dene the spectral zeta function ?T (?, z) attached to T of Hurwitz's
type by
?
X
?T (w, z) :=
(?j + z)?w .
j=0
We assume that the series converges absolutely in some right half w-plane (uniformly
for z on any compact set) and can be continued meromorphically to a region
containing w = 1 ? r for r ? N. Moreover, we assume that ?T (w, z) is holomorphic
at w = 1 ? r. In this case, we dene a higher depth determinant of T of depth r by
?
?T (w, z)
.
Detr (T + z) := exp ?
?w
w=1?r
This can be considered as a determinant analogue of the Milnor gamma function
?
?(w, z)|w=1?r )
?w
P?
?w
studied in [7] (see also [4]). Here ?(w, z) :=
is the Hurwitz zeta
n=0 (n + z)
function. Note that det(T + z) := Det1 (T + z) (with z = 0) gives the usual
normalized determinant of T (see, e.g., [9]).
?r (z) := exp(
1 This
1790
work is partially supported by Grant-in-Aid for JSPS Fellows No.19002485
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
The aim of the present paper is to evaluate the higher depth determinants when
T is the Laplacian ?? on the compact Riemann surface R = ?\H, where H is the
complex upper half plane with the standard Poincare metric and ? is a discrete,
co-compact torsion-free subgroup of SL2 (R). We show the following theorem, which
gives a generalization of the result for r = 1 in [11] (see also [8]).
Theorem 1.1
[5]
The higher depth determinants
Detr (?? ? s(1 ? s))
can be
explicitly expressed as a product of the multiple gamma functions and Milnor-Selberg
zeta functions.
This is a summary article; readers who are interested in this topic can nd the
detailed proof of the above result in the forthcoming paper [5]. See also [13] for
explicit calculations of the higher depth determinants of the Laplacian on spheres
in higher dimensions.
џ 2. A product expression of Detr (?? ? s(1 ? s))
We rst recall the Selberg trace formula for the Riemann surface R = ?\H of
genus g > 2. Let f be a function whose Fourier transform
Z ?
fb(r) :=
f (x)e?irx dx
??
satises the conditions fb(?r) = fb(r), fb is holomorphic in the band |Im r| < ? + 1/2
and fb(r) = O(|r|?2?? ) as |r| ? ? for some ? > 0. Then, the formula reads
?
X
mj fb(rj ) =
j=0
log N (?? )
f (log N (?))
? N (?)?1/2
X
??Hyp(?)
N (?)1/2
Z
?
+ (g ? 1)
fb(r) r tanh(?r) dr.
(2.1)
??
Here mj is the multiplicity of ?j (j > 0), rj is the number determined by ?j = 14 + rj2
(rj > 0 if rj ? R and ?irj > 0 otherwise), Hyp(?) (resp. Prim (?)) is the set of all
hyperbolic (resp. primitive) conjugacy classes in ?, N (?) is the square of the larger
eigenvalue of ? ? Hyp(?) and, for ? ? Hyp(?), ?? ? Prim (?) is the unique element
satisfying ? = ??k for some k > 1.
Suppose Re s > 1/2 and Re w > r. Write t = s ? 1/2. Let
f (x) := ?
x w?r+1/2
1
Kw?r+1/2 (tx),
? ?(w + 1 ? r) 2t
where ?(x) is the classical gamma function and K? (x) is the K -Bessel function.
Then, taking this f as a test function of the trace formula (2.1) and noticing that
1791
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
fb(r) = (r2 + t2 )?w+r?1 , we have
??? w + 1 ? r, ?s(1 ? s) = Ir w, t +
r?1 X
r ? 1 2(r?1?`) (2`+1)
+ (g ? 1)
t
J
(w, t),
`
`=0
(2.2)
where
log N (?? )
Ar (w, a; ?),
? N (?)?1/2
??Hyp(?)
w?r+1/2
log N (?)
1
Ar (w, a; ?) := ?
Kw?r+1/2 a log N (?)
2a
? ?(w ? r + 1)
Ir (w, a) :=
X
N (?)1/2
and
J
(m)
Z
?
(w, a) :=
(x2 + a2 )?w xm tanh(?x) dx.
??
One can show that both Ir (w, a) and J (m) (w, a) are continued meromorphically to
C as a function of w and are in particular holomorphic at w = 0. Hence, taking the
derivatives at w = 0 of the both hands side of (2.2), we have
Detr ?? ? s(1 ? s) = ?r (s)g?1 Z?,r (s),
(2.3)
where
r?1
Y
(r?1)t2(r?1?`)
?
`
,
exp ? J (2`+1) (w, t)
?w
w=0
`=0
?
.
Z?,r (s) : = exp ? Ir (w, t)
?w
w=0
?r (s) : =
(2.4)
(2.5)
In the subsequence sections, we calculate the gamma factor ?r (s) (Theorem 3.1)
and the zeta factor Z?,r (s) (Theorem 4.1), respectively. As a consequence, our
main result (Theorem 1.1) follows immediately from the equation (2.3).
џ 3. Gamma factor
To evaluate ?r (s), we here recall the Barnes multiple gamma functions. Let
X
1
?n (w, z) :=
,
Re w > n,
(m1 + . . . + mn + z)w
m ,...,m >0
1
n
be the Barnes multiple zeta function [2]. It is known that ?n (w, z) can be continued
meromorphically to C with possible simple poles at w = 1, 2, . . . , n. Then, the Barnes
multiple gamma function ?n (z) is dened by
?
?n (z) := exp
?n (w, z)
.
?w
w=0
1792
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
?
Note that ?1 (z) = ?(z)/ 2? from the Lerch formula [6]
?
?(z)
?(w, z)w=0 = log ? .
?w
2?
One can evaluate the integral J (m) (w, a) by using the residue theorem and its
derivative at w = 0 in terms of (the logarithm of) the Barnes multiple gamma
function. Consequently, together with the formula (2.4), we have the following
expression of ?r (s);
Theorem 3.1
where
?r,j (t)
t = s ? 1/2. Then we have
Y
2r
(2r)!!
2r
·
t
?j (s)?r,j (t) ,
?r (s) = exp ? 2
r (2r ? 1)!!
j=1
Write
is the even polynomial given by
r?1 X
r?1
1
`
?r,j (t) := 4
(?1) c2`+2,j
t2(r?1?`) .
`
2
j?1
`=[
Here
[x]
2
]
denotes the largest integer not exceeding
dened by
(T + z)
x
cr,j (z)
and
is the polynomial
r
X
T +j?1
=
cr,j (z)
.
j?1
j=1
r?1
Example 3.2 Write t = s ? 1/2. Then it holds that
2
?1 (s) = e?2t ?1 (s)?2 ?2 (s)4 ,
2+ 1
2
2 4
?2 (s) = e? 3 t ?1 (s)?2t
16 6
1
2 ?2t4
?3 (s) = e? 45 t ?1 (s)? 8 +t
2 ?13
?2 (s)4t
?2 (s)
?3 (s)36 ?4 (s)?24 ,
121
?26t2 +4t4
4
2
?3 (s)?330+72t
2
Ч ?4 (s)1020?48t ?5 (s)?1200 ?6 (s)480 .
Remark 3.3
The function ?r (s) can be also expressed in terms of the Vigneras
multiple gamma functions Gn (z) [12], see also [1] for the case n = 2, which are
characterized by a generalization of the Bohr-Mollerup theorem. We remark that
?n (z) and Gn (z) are essentially equal (see, more precisely, [10]).
џ 4. Zeta factor
We next evaluate Z?,r (s). To do that, we introduce a Milnor-Selberg zeta function
of depth m by the following Euler product:
(m)
Z? (s)
(m)
Z? (s) :=
Y
?
Y
Hm N (P )?s?n
(log N (P ))?m+1
,
Re s > 1.
(4.1)
P ?Prim (?) n=0
1793
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
Here Hm (z) := exp(?Lim (z)) with
?
X
zk
Lim (z) :=
km
k=1
being the polylogarithm function. Notice that, since Li1 (z) = ? log (1 ? z), this gives
the Selberg zeta function
Y
Z? (s) :=
?
Y
1 ? N (P )?s?n ,
Re s > 1.
P ?Prim (?) n=0
We remark that, since
d
1
Lim (z) = Lim?1 (z),
dz
z
(m)
the Milnor-Selberg zeta functions Z? (s) satisfy the following dierential ladder
relation;
dm?2
dm?1
(m)
(m?1)
log
Z
(s)
=
?
log Z?
(s) = . . . = (?1)m?1 log Z? (s).
?
dsm?1
dsm?2
This shows that the Milnor-Selberg zeta function of depth m is essentially given by
the (m ? 1)-th iterated integrals of the logarithm of the Selberg zeta function Z? (s).
Therefore, since Z? (s) has zeros at s = ?k for k = ?1, 0, 1, 2, . . . and (1/2) ± irj for
j = 1, 2, . . ., it is in general a multi-valued function.
Since the K -Bessel function K? (x) is analytic with respect to the variable ? , one
can see that the functions Ar (w, a; ?) and hence Ir (w, a) are holomorphic at w = 0.
Moreover, using the asymptotic formulas
Kw?r+1/2 (a log N (?)) = K?r+1/2 a log N (?) + O(w),
w?r+1/2 ?r+1/2
log N (?)
log N (?)
=
+ O(w),
2a
2a
1
= (?1)r?1 (r ? 1)! w + O(w2 )
?(w ? r + 1)
as w ? 0 together with the well-known formula, see, e. g., [3],
r?1
? 1/2
X
(r + m ? 1)!
?y
e
(2y)?m
K?r+1/2 (y) = Kr?1/2 (y) =
,
2y
m!(r ? m ? 1)!
m=0
we have
r?1 X
?
r?1
r?1
(r + m ? 1)! Ч
Ir (w, a)
= (?1)
?w
m
w=0
m=0
1
r?1?m
Ч (2a)
z? a + , r + m ,
2
1794
(4.2)
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
where
z? (s, m) :=
X
??Hyp(?)
log N (?? )
N (?)?s+1/2
·
.
N (?)1/2 ? N (?)?1/2 (log N (?))m
By a straightforward calculation, one can show that
(m)
log Z? (s) = ?z? (s, m).
Therefore, together with the formulas (2.5) and (4.2), we obtain the following
Theorem 4.1
Write
Z?,r (s) =
t = s ? 1/2.
r?1
Y
Then we have
!(?1)r?1
r?1
r?1?m
(r+m)
Z?
(s)( m )(r+m?1)!(2t)
,
Re s > 1.
m=0
Example 4.2 Write t = s ? 21 . Then it holds that
(1)
Z?,1 (s) = Z? (s) = Z? (s),
(2)
(3)
Z?,2 (s) = Z? (s)?2t Z? (s)?2 ,
(3)
2
(4)
(5)
Z?,3 (s) = Z? (s)8t Z? (s)24t Z? (s)24 .
Remark 4.3
One can see that the function Z?,1 (s) is an entire function because
(1)
the Selberg zeta function Z? (s) is (notice that Z?,1 (s) = Z? (s) = Z? (s)). However,
for r > 2, it has not been claried whether Z?,r (s), which is written as a product of
(m)
Z? (s), can be (analytically) continued to a single-valued function (recall that the
(m)
Milnor-Selberg zeta function Z? (s) is in general a multi-valued function).
References
1. E. W. Barnes. The theory of the G-function, Quart. J. Math., 1899, vol. 31,
264314.
2. E. W. Barnes. On the theory of the multiple gamma functions, Trans.
Cambridge Philos. Soc., 1904, vol. 19, 374425.
3. A. Erdelyi, W. Magnus, F. Oberthettinger and F. G. Tricomi. Higher Transcendental Functions, McGraw-Hill, New York, 1953.
4. N. Kurokawa, H. Ochiai and M. Wakayama. Milnor's multiple gamma
functions, J. Ramanujan Math. Soc., 2002, vol. 21, 153167.
5. N. Kurokawa, M. Wakayama and Y. Yamasaki. Higher depth determinants of
Laplacians and Milnor-Selberg zeta functions, preprint, 2008.
1795
Вестник ТГУ, т. 16, вып. 6, ч. 2, 2011
e Akad.,
6. M. Lerch. Dalsi studie v oboru Malmstenovsk
ych rad, Rozpravy Cesk
1894, vol. 3, No. 28, 161.
7. J. Milnor. On polylogarithms, Hurwitz zeta functions, and the Kubert
identities, Enseignement Mathematique, 1983, vol. 29, 281322.
8. P. Sarnak. Determinants of Laplacians, Commun. Math. Phys., 1987, vol. 110,
113120.
9. C. Soule, D. Abramovich, J.-F. Burnol and J. Kramer. Lectures on Arakelov
geometry, Cambridge Studies in Advanced Mathematics, 33. Cambridge University
Press, Cambridge, 1992.
10. H. M. Srivastava and J. Choi. Series associated with the zeta and related
functions, Kluwer Academic Publishers, Dordrecht, 2001.
11. A. Voros. Spectral functions, special functions and the Selberg zeta functions,
Commun. Math. Phys., 1987, vol. 110, 439465.
12. M. F. Vigneras. L'equation fonctionelle de la fonction zeta de Selberg de
groupe modulaire PSL(2, Z), Asterisque, 1979, vol. 61, 235249.
13. Y. Yamasaki. Higher depth determinants of the Laplacian on n-sphere.
Preprint, 2008.
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