Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps

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by  Joshua S. Friedman

SUPERZETA FUNCTIONS, REGULARIZED PRODUCTS, AND
THE SELBERG ZETA FUNCTION ON HYPERBOLIC
MANIFOLDS WITH CUSPS
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2 JOSHUAS.FRIEDMAN,JAYJORGENSON,ANDLEJLASMAJLOVIC´

n
a Abstract. LetΛ={λk}denoteasequenceofcomplexnumbersandassume
J that that the counting function #{λk ∈ Λ : |λk| < T} = O(Tn) for some
4 integer n. From Hadamard’s theorem, we can construct an entire function f
2 oforderatmostnsuchthatΛisthedivisorf. Inthisarticleweprove,under
reasonablygeneralconditions,thatthesuperzetafunctionZf(s,z)associated

] to Λ admits ameromorphiccontinuation. Furthermore, wedescribethe rela-
T tion between the regularized product of the sequence z−Λ and the function
N f as constructed as a Weierstrass product. In the case f admits a Dirichlet
seriesexpansioninsomerighthalf-plane,wederivethemeromorphiccontinu-
h. ationinsofZf(s,z)asanintegraltransformoff′/f. Weapplytheseresults
t toobtainsuperzetaproductevaluationsofSelbergzetafunctionassociatedto
a
finitevolumehyperbolicmanifoldswithcusps.
m
[

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v 1. Introduction
9
1.1. Background discussion. The zeta-regularized product approach to define
6
8 the product of an infinite sequence of complex numbers is, at this point, a well-
6 acceptedmathematical concept. Let Λ= λ denote a sequence of complex num-
k
{ }
0 bers and assume that that the counting function # λ Λ : λ < T = O(Tn)
k k
. { ∈ | | }
1 for some integer n. For any complex number s with Re(s)>n, the function
0
ζ (s)= λ−s
7 Λ k
1 λk∈XΛ,λk6=0
:
v is convergent. Under additional conditions, ζΛ(s) admits a meromorphic continua-
i tion to a region which includes s =0 and is holomorphic at s =0. In such a case,
X
one defines the zeta-regularized product of Λ, which is denoted by det∗(Λ), to be
ar det∗(Λ) = exp(−ζΛ′(0)) where the prime denotes a derivative with respect to the
variables. The use ofthe termdeterminant stems fromexample when Λ is a finite
sequence of non-zero numbers associated to a finite dimensional linear operator T
in which case exp( ζ′(0)) is, indeed, the determinant of T. In general, the use of
− Λ
the asterisk signifies that one considers only the non-zero elements of Λ.
As stated, the zeta-regularized product of Λ requires that Λ satisfies additional
conditions which are listed as generalaxioms in [11]. One of the conditions is that
the associated theta function θ (t) = e−λkt is defined for real and positive
Λ
λXk∈Λ
2010 Mathematics Subject Classification. Primary11M36.
First-named author: The views expressed in this article are the author’s own and not those
of the U.S.Merchant MarineAcademy, the MaritimeAdministration, the Department of Trans-
portation,ortheUnitedStates government.
Second-named author’sresearchissupportedbyNSFandPSC-CUNYgrants.
1
2 JOSHUAS.FRIEDMAN,JAYJORGENSON,ANDLEJLASMAJLOVIC´
valuesoftandadmitsanasymptoticexpansionastapproacheszero. Thiscondition
allows one to compute ζ (s) as the Mellin transform of θ (t). If Λ is the sequence
Λ Λ
of eigenvalues of some Laplacian operator ∆ acting of a Hilbert space associated
to compact Riemannian manifolds, then one views det∗(Λ) as the determinant of
∆ and one writes det∗∆=det∗(Λ).
In [23] the authors consider, among other ideas, the zeta-regularizedproduct of
the sequence of eigenvalues associatedto the Laplacians acting on smooth sections
of flat line bundles over genus one curves, when equipped with a unit volume flat
metric,andhighergenusRiemannsurfaces,whenequippedwithhyperbolicmetrics
of constant negative curvature. In the genus one case, the evaluation of the deter-
minant of the Laplacian amounts to Kronecker’s second limit formula, resulting in
the evaluation of determinant of the Laplacian in terms of Riemann’s theta func-
tion. For higher genus surfaces, the determinant of the Laplacian is expressed in
terms of Selberg’s zeta function. Far-reaching applications of determinants of the
Laplacian, and various sums known as analytic torsion, provide the underpinnings
of the analytic aspect of Arakelov theory.
1.2. Our main results. If we continue to consider the setting of Laplacians on
finite volume Riemann surfaces, we note that the zeta-regularized approach to
defining a determinant of the Laplacian does not extend to the setting when the
surface admits cusps. Articles exist where the authors propose means to define
determinantsof the Laplacianin the non-compactsetting by regularizingthe trace
of the heat kernel in some sense; see, for example, [12], [13], [14] and [21]. The
purpose of the present article is to provide another approach to the problem. The
point of view we consider can be explained through the following example. Let M
denoteafinitevolumehyperbolicRiemannsurface,andlet∆denotetheassociated
Laplacian which acts on the space of smooth functions on M. If M is compact,
the spectrum of ∆ consists of discrete eigenvalues. If M is not compact, then the
correspondingspectrumhasdiscreteeigenvalues,whosenatureisthesubjectofthe
Phillips-Sarnak philosophy [22], as well as a continuous spectrum with associated
measuredµwhichisabsolutelycontinuouswithrespecttoLebesguemeasureonR.
Furthermore, one can write dµ(r) = φ′/φ(1/2+ir)dr where dr denotes Lebesgue
measure and φ(s) is a meromorphic function of s C but whose restriction to the
∈
line Re(s) = 1/2 appears in the evaluation of the spectral measure. The set Λ to
be constructedconsistsof the poles ofthe function φ(s) for all s C together with
additional points in C associated to the L2 spectrum, appropria∈tely computed in
the parameter r.
Once the set Λ has been determined, then the regularized product we define
follows the super-zeta function construction which was systematically developed
by Voros; see [24], [25] and [26]. In vague terms, which will be clarified below,
one defines (s,z) = (z λ )−s for s C with Re(s) sufficiently large and
f k
Z − ∈
z CsuitablyrestrictedλX.k∈TΛhesuperzetaregularizedproductofz Λisobtainedby
∈ −
provingthe meromorphiccontinuationin s andemployinga specialvalue ats=0.
The purpose of this paper is to establish general circumstances under which
the proposed construction is valid. In section 2 we prove that for a general entire
function f of finite order, one can define the superzeta regularized product of its
zeros. Furthermore,we establisha preciserelationbetweenthe Hadamardproduct
representation of f and exp( ′(0,z)) where the prime denotes a derivative in s.
−Zf
SUPERZETA FUNCTIONS 3
In section 3 we present an example of the results from section 2 taking f to be the
Selberg zeta function associated to any co-finite Kleinian group. In section 4 we
generalize results of section 2, by assuming that f is a finite order meromorphic
function which admits a general Dirichlet series representation. We then prove,
in Theorem 4.1, that one has the meromorphic continuation of the corresponding
superzetafunction. Section5yieldsanexampleofthe settingofsection4obtained
byconsideringthe Selbergzeta function associatedtoany finite volume hyperbolic
manifold of arbitrary dimension.
To summarize, the main results of this article combine to prove the superzeta
regularizedproductforanymeromorphicfunctionwhichadmitsageneralDirichlet
seriesrepresentation. Theresultismoregeneralthantheso-called“laddertheorem”
from [10] which requires a functional equation.
Finally, let us note that circumstances may occur when a sequence Λ may arise
from an operator. In [5] we took Λ to be the divisor of the Selberg zeta function
for a finite volume, non-compact hyperbolic Riemann surface. In this case, it was
discussedin[5]howΛcanbeviewedintermsoftheLax-Phillipsscatteringoperator
[19], so then we obtain a construction of what could be viewed as the determinant
of this operator.
2. Zeta regularization of entire function
LetR− =( ,0]bethenon-positiverealnumbers. Let yk k∈Nbethesequence
−∞ { }
of zeros of an entire function f of order κ 1, repeated with their multiplicities.
≥
Let
X = z C (z y ) / R− for all y .
f k k
{ ∈ | − ∈ }
For z X , and s C (where convergent)consider the series
f
∈ ∈
∞
(2.1) (s,z)= (z y )−s,
f k
Z −
k=1
X
where the complex exponent is defined using the principal branch of the logarithm
with argz ( π,π) in the cut plane C R−.
∈ − \
Since f is of order κ, (s,z) converges absolutely for Re(s) > κ. The series
f
Z
(s,z) is called the zeta function associated to the zeros of f, or the simply the
f
Z
superzeta function of f.
If (s,z) has a meromorphic continuationwhich is regularat s=0, we define,
f
Z
for z X the zeta regularized product associated to f as
f
∈
d
D (z)=exp (s,z) .
f −ds Zf |s=0
(cid:18) (cid:19)
Let m = κ , where κ denotes the largest integer less than or equal to κ.
⌊ ⌋ ⌊ ⌋
Hadamard’s product formula allows us to write
∞ z z zm
(2.2) f(z)=∆ (z)=eg(z)zr 1 exp +...+ ,
f − y y my m
k=1(cid:18)(cid:18) k(cid:19) (cid:20) k k (cid:21)(cid:19)
Y
where g(z) is a polynomial of degree m or less, r 0 is the order of the eventual
≥
zero of f at z = 0, and the other zeros y are listed with multiplicity. A simple
k
calculation shows that when z X ,
f
∈
( 1)m
(2.3) (m+1,z)= − (log∆ (z))(m+1).
f f
Z m!
4 JOSHUAS.FRIEDMAN,JAYJORGENSON,ANDLEJLASMAJLOVIC´
ThefollowingpropositionisduetoVoros([24],[26],[27]). Wealsogiveadifferent
proof.
Proposition1. Let f be an entirefunction of order κ 1, andfor k N, let y be
k
≥ ∈
the sequence of zeros of f. Let ∆ (z) denote the Hadamard product representation
f
of f. Assume that for n>m= κ we have the following asymptotic expansion:
⌊ ⌋
m m n−1
(2.4) log∆f(z)= ajzj(logz Hj)+ bjzj + akzµk +hn(z),
−
j=0 j=0 k=1
X X X
j e
where H =0, H = (1/l), for j 1, 1>µ >...>µ , and h (z) is
0 j l=1 ≥ 1 n →−∞ n
a sequence of holomorphic functions in the sector argz < θ < π, (θ > 0) such
P | |
that h(nj)(z)=O(z µn−j), as z in the above sector, for all integers j 0.
| | | |→∞ ≥
Then, for all z X , the superzeta function (s,z) has a meromorphic con-
f f
∈ Z
tinuation to the half-plane Re(s) < κ which is regular at s = 0. Furthermore, the
zeta regularized product D (z) associated to (s,z) is related to ∆ (z) through
f f f
Z
the formula
(2.5) Df(z)=e−(Pmj=0bjzj)∆f(z)
which also provides analytic continuation of D (z) from X to the whole complex
f f
z plane.
−
Proof. For any z X , the series
f
∈
∞
(2.6) (m+1,z+y)= (z+y y )−(m+1)
f k
Z −
k=1
X
converges absolutely and uniformly for y (0, ). Furthermore, application of [8,
∈ ∞
Formula 3.194.3],with µ=m+1 s, ν =m+1 and β =(z y )−1 yields, for all
k
− −
y ,
k
∞
ym−sdy 1
= (z y )−sΓ(m+1 s)Γ(s).
(z+y y )m+1 m! − k −
k
Z −
0
Absolute and uniform convergence of the series (2.6) for Re(s)>κ implies that
m! ∞
(2.7) (s,z)= (m+1,z+y)ym−sdy
f f
Z Γ(m+1 s)Γ(s) Z
− Z0
( 1)m ∞
= − (log∆ (z+y))(m+1)ym−sdy,
f
Γ(m+1 s)Γ(s)
− Z0
for κ<Re(s)<m+1.
Next, we use (2.7) together with (2.4) in order to get the meromorphic con-
tinuation of Z (s,z) to the half plane Re(s) < m+1. We start with (2.3) and
f
differentiate Equation (2.4) (m+1) times to get
m ( 1)m−jj!(m j)!a n−1a µ (µ 1) ... (µ m)
(log∆ (z))(m+1) = − − j + k k k− · · k−
f (z+y)(m+1−j) (z+y)m+1−µk
j=0 k=1
X e X
+h(m+1)(z+y),
n
for any n>m.
SUPERZETA FUNCTIONS 5
Since µ , for an arbitrary µ<0 there exists k such that µ µ for all
k 0 k
ց−∞ ≤
k k , hence we may write
0
≥
m ( 1)m−jj!(m j)!a
(log∆ (z))(m+1)ym+1 =ym+1 − − j
f  (z+y)(m+1−j)
j=0
X e
n−1 
a µ (µ 1) ... (µ m)
k k k k
+ − · · − +g (z+y),
(z+y)m+1−µk ! µ
k=1
X
where g (z+y)=ym+1h(m+1)(z+y).
µ k0
Note that
(2.8) g (z+y)=O(yµ) as y , and g (z+y)=O(ym+1) as y 0.
µ µ
→∞ ց
Application of [8, Formula 3.194.3]yields
(2.9)
( 1)m ∞ m Γ(s j)
− (log∆ (z+y))(m+1)ym−sdy = ( 1)jj!a − zj−s
f j
Γ(m+1 s)Γ(s) − Γ(s)
− Z0 j=0
X
k0−1 Γ(s µ ) ( 1)m ∞ e
ak − k zµk−s+ − gµ(z+y)y−s−1dy.
− Γ(s)Γ( µ ) Γ(m+1 s)Γ(s)
k=1 − k − Z0
X
The integral on the right hand side of (2.9) is the Mellin transformof the function
g .By (2.8)this integralrepresentsaholomorphicfunction ins foralls inthe half
µ
stripµ<Re(s)<m+1. Theothertermsonthe righthandsideof (2.9)aremero-
morphic in s, hence, by (2.7), the right-hand side of (2.9) provides meromorphic
continuationofintegral ∞Z (m+1,z+y)y2−sdyfromthestripκ<Re(s)<m+1
0 f
to the strip µ < Re(s) < m+1. Since µ < 0 was chosen arbitrarily, we can let
R
µ andobtainthemeromorphiccontinuationofthisintegraltothehalfplane
→−∞
Re(s)<m+1.
Formula (2.9), together with (2.7), now yields the following representation of
(s,z), for an arbitrary,fixed z X , valid in the half plane µ<Re(s)<m+1:
f f
Z ∈
m Γ(s j) k0−1 Γ(s µ )
(2.10) f(s,z)= ( 1)jj!aj − zj−s ak − k zµk−s
Z − Γ(s) − Γ(s)Γ( µ )
k
j=0 k=1 −
X X
e + (−1)m ∞h(m+1)(z+y)ym−sdy.
Γ(m+1 s)Γ(s) k0
− Z0
From the decay properties of h(m+1)(z+y), it follows that (s,z) is holomorphic
k0 Zf
ats=0. Furthermoresince 1 has a zeroats=0, the derivativeofthe lastterm
Γ(s)
in (2.10) is equal to
d 1 ( 1)m ∞ ( 1)m ∞
− h(m+1)(z+y)ymdy = − h(m+1)(z+y)y2dy
dsΓ(s) Γ(m+1) k0 m! k0
(cid:18) (cid:12)s=0(cid:19) Z0 Z0
(cid:12)(cid:12)(cid:12) =−hk0(z),
6 JOSHUAS.FRIEDMAN,JAYJORGENSON,ANDLEJLASMAJLOVIC´
where the last equality is obtained from integration by parts m times, and using
thedecayofh (z+y)anditsderivativesasy + ,forµ <0.Moreover,since
k0 → ∞ k0
d Γ(s µ ) d 1
k
− =Γ( µ ) =Γ( µ ),
k k
ds Γ(s) − · dsΓ(s) −
(cid:12)s=0 (cid:12)s=0
(cid:12) (cid:12)
elementary computations yie(cid:12)ld that (cid:12)
(cid:12) (cid:12)
d k0−1 Γ(s µ ) k0−1
ak − k zµk−s = akzµk.
ds Γ(s)Γ( µk) !(cid:12)
Xk=1 − (cid:12)s=0 kX=1
(cid:12)
This shows that (cid:12)
(cid:12)
(2.11)
d d m Γ(s j) k0−1
− dsZf(s,z) =− ds (−1)jj!aj Γ(−s) zj−s(cid:12) + akzµk+hk0(z),
(cid:12)s=0 j=0 (cid:12) k=1
(cid:12) X (cid:12)s=0 X
(cid:12)  e (cid:12)
for z in the se(cid:12)ctor argz <θ <π, (θ >0). (cid:12)
| | (cid:12)
Now, for j 1,...,m one has
∈{ }
j
d Γ(s j) d Γ(s j)
− zj−s =( logz)zj (0 k)−1+zj − .
ds Γ(s) − − ds Γ(s)
(cid:18) (cid:19)(cid:12)s=0 k=1 (cid:12)s=0
(cid:12) Y (cid:12)
(cid:12) (cid:12)
A straightforwardcompu(cid:12)tation shows that (cid:12)
d Γ(s j) d j ( 1)j j 1 ( 1)j
− = (s k)−1 = − =H − .
j
Therefodrse, Γ(s) (cid:12)(cid:12)(cid:12)(cid:12)s=0 dskY=1 − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s=0 j! Xk=1k · j!
d Γ(s j)
( 1)jj!a − zj−s =a ( logz)zj +a zjH ,
j j j j
ds − Γ(s) −
(cid:18) (cid:19)(cid:12)s=0
(cid:12)
for j 1,...,m and heence (cid:12) e e
∈{ } (cid:12)
m m
d Γ(s j)
( 1)jj!a − zj−s = a zj(logz H ).
j j j
− ds − Γ(s) (cid:12) −
j=0 (cid:12) j=0
X (cid:12)s=0 X
 e (cid:12) e
Now,(2.5)followsfromequation(2.11)and(cid:12) uniquenessofanalyticcontinuation.
(cid:12)
(cid:3)
3. Superzeta functions constructed from confinite Kleinian groups
Let Γ be a cofinite Kleinian group. Suppose T Γ is loxodromic. Then T is
∈
conjugate in PSL(2,C) to a unique element of the form a(T) 0 such that
0 a(T)−1
a(T) C has a(T) >1. Let N(T):= a(T)2, and let (T(cid:16)) denote the(cid:17)centralizer
∈ | | | | C
of T in Γ. There exists a (primitive) loxodromic element T , and a finite cyclic
0
ellipticsubgroup (T)oforderm(T),generatedbyanelementE suchthat (T)=
T
T (T), wheEre T = Tn n Z . C
h 0Tih×eEelliptic elemhen0tiE {is0co|nju∈gate}in PSL(2,C) to an element of the form
T
ζ(T0) 0 where here ζ(T ) is a primitive 2m(T)-th root of unity.
0 ζ(T0)−1 0
(cid:16) (cid:17)
SUPERZETA FUNCTIONS 7
Definition 2. For Re(s)>1 the Selberg zeta-function Z(s) is defined by
Z(s):= 1 a(T )−2ka(T )−2lN(T )−s−1 .
0 0 0
−
{TY0}∈R l,Yk≥0 (cid:16) (cid:17)
c(T,l,k)=1
Here the product with respect to T extends over a maximal reduced system of
0
R
Γ-conjugacy classes of primitive loxodromic elements of Γ. The system is called
R
reduced if no two of its elements have representatives with the same centralizer1.
The function c(T,l,k) is defined by c(T,l,k)=ζ(T )2lζ(T )−2k.
0 0
Let ζ P1 be cusp of Γ, Γ := γ Γ γζ = ζ , and let Γ′ be the maximal
∈ ζ { ∈ | } ζ
torsion-freeparabolicsubgroupofΓ . The possiblevalues for the index of[Γ :Γ′]
ζ ζ ζ
are 1,2,3,4, and 6.
Proposition3([3]). LetΓbecofinitewithoneclassofcuspat . If[Γ :Γ′ ]=1
∞ ∞ ∞
or [Γ :Γ′ ]=2, then Z(s) is a meromorphic function. Furthermore, the zeros of
∞ ∞
the Selberg zeta-function are:
(1) zeros at the points s on the line Re(s) = 0 and on the interval [ 1,1].
j
± −
Each point s is related to an eigenvalue λ of the discrete spectrum of (the
j j
self-adjoint extension) of Laplacian ∆ by 1 s2 = λ . The multiplicity
− − j j
of each s > 0, m(s ), is equal to the multiplicity of the corresponding
j j
eigenvaluem(λ ).If s <0happenstoalsobeazeroofφ(s)ofmultiplicity
j j
−
q( s ) then m( s ) = m(λ ) q( s ). Here φ(s) is determinant of the
j j j j
− − − −
automorphic scattering matrix (in this case, it is a scalar since we assume
the number of cusps is one).
(2) zeros at the points ρ , that are poles of φ(s), which lie in the half-plane
j
Re(s)<0. The multiplicity of each ρ is equal its multiplicity, as a pole, of
j
φ(s).
(3) if [Γ :Γ′ ]=1 then zeros at the points s=Z with multiplicity 1; at the
∞ ∞ <0
points=0itcouldbeapoleorazero, withmultiplicity 1(φ(0) 1)+2m(1),
2 −
where m(1) denotes the multiplicity of the possible eigenvalue λ=1.
(4) if [Γ : Γ′ ] = 2 then zeros at the points s = 1, 3, 5,..., with multi-
∞ ∞ − − −
plicity 1; at the point s = 0 it could be a pole or a zero with multiplicity
1(φ(0) 1)+2m(1).
2 −
Let ( ∗ ∗) denote the representatives of Γ′ Γ / Γ′ . For Re(s) > 1, the
cn d ∈ R ∞\ ∞
scattering matrix φ(s) can be written as a Dirichlet series
π m(c )
φ(s)= 0 c −2−2s(1+ q−2−2s),
[Γ :Γ′ ] s | 0| n
∞ ∞ |P| n≥1
X
where q = c /c >1, is the co-area of the lattice associated to Γ , c is the
n n 0 ∞ 0
| | |P|
minimal (non-zero) modulus lower left entry for Γ, and m(c ) counts the number
0
of times c is present as a representative in . See [2, p. 111 and p.234 Eq. 1.10])
0
R
for more details.
3.1. Thecase[Γ :Γ′ ]=1. When[Γ :Γ′ ]=1,wedefineZ+(s)=Z(s)/Γ(s),
∞ ∞ ∞ ∞ 1
andlet +(s,z)beitssuperzetafunction. FromtheasymptoticversionofStirling’s
Z1
1See[2]section5.4formoredetails
8 JOSHUAS.FRIEDMAN,JAYJORGENSON,ANDLEJLASMAJLOVIC´
formula and from the product expansion of Z(s), it follows that as Re(s) + ,
→ ∞
we have that
1 1
logZ+(s) (logs 0) 1s(logs 1) log2π.
1 ∼ 2 − − − − 2
Hence applying Proposition 1 to the finite-order entire function Z+(s), we obtain
1
the regularizedproduct of the zeros of Z+(s) as D (s)=√2πZ+(s).
1 Z+ 1
1
Next let Z−(s)=φ(s)Z+(s). Hence it follows that as Re(s) + ,
1 1 → ∞
1 1 πm(c )
logZ−(s) (logs 0) 1s(logs 1) log2π+log 0 slog(c 2 ).
1 ∼ 2 − − − −2 c 2 − | 0| |P|
(cid:18) | 0| (cid:19)
Once again, applying Proposition 1 we obtain that
c 2s s c 2
DZ1−(s)= | 0|πm|P(c|0|)0| √2πZ1−(s).
Recalling that Z−(s) = φ(s)Z+(s) we obtain a formula for φ(s) as a quotient of
1 1
regularizeddeterminants,
φ(s)= DZ1−(s) πm(c0)
D (s) c 2s+1 s
Z1+ | 0| |P|
from which one can obtain a formula for the central value of φ(s); see [6].
3.2. The case [Γ : Γ′ ]= 2. When [Γ : Γ′ ]= 2, (this is the case e.g. for the
∞ ∞ ∞ ∞
Picard group) define
sZ(s)
Z+(s)= ,
2 Γ s−1
2
and let Z2+(s,z) be its superzeta function. (cid:0) (cid:1)
The asymptotic expansion of logΓ(s−1) can be computed as
2
s 1 1 s 1 1
logΓ( − ) slogs slog2 logs+ log2π+log2.
2 ∼ 2 − 2 − 2 − 2
Hence from the product expansion of Z(s), it follows that as Re(s) + ,
→ ∞
1 s 1 1
logZ+(s) 2logs slogs+ + slog2 log2π log2
2 ∼ − 2 2 2 − 2 −
and can be rewritten in the form required by Proposition1
1 log2 1
logZ+(s) 2(logs 0) s(logs 1)+ s log2π log2.
2 ∼ − − 2 − 2 − 2 −
Hence
DZ+(s)=√π2(3−2s)Z1+(s).
2
If we define Z−(s)=φ(s)Z+(s) then we obtain
2 2
DZ2−(s)= |c0|π2sm|P(c|s0|)c0|2√π2(5−2s)φ(s)Z2+(s).
Finally we obtain
φ(s)= DZ2−(s) πm(c0) .
D (s)2c 2s+2 s
Z2+ | 0| |P|
SUPERZETA FUNCTIONS 9
4. Zeta regularization of zeta-type functions through integral
representation
Inthissectionweassumethatf isameromorphicfunctionoffiniteorderκsuch
that logf possesses a representation as a generalized Dirichlet series
∞
c
n
(4.1) logf(z)=
qs
n=1 n
X
convergingabsolutelyanduniformly inanyhalfplane Re(s) σ+ǫ,foranyǫ>0.
≥
Here q denotes an increasing sequence of real numbers with q > 1 and c are
n 1 n
{ }
complex numbers. For such a function f we say it is a zeta-type function.
We denote by N the set ofzerosoff andby P the setof poles off anddefine
f f
X = z C (z z ) / R− for all z N P .
f k k f f
{ ∈ | − ∈ ∈ ∪ }
For z X , and s C with Re(s)>κ consider the function
f
∈ ∈
(4.2) (s,z):= ord(ρ)(z ρ)−s ord(ρ)(z ρ)−s := N(s,z) P(s,z),
Zf − − − Zf −Zf
ρX∈Nf ρX∈Pf
where ord(ρ) denotes the order of a zero or pole. We call (s,z) the superzeta
f
Z
function associated to the meromorphic function f. Note that both series in (4.2)
are absolutely convergentfor Re(s)>κ.
Denote byWN andWP Weierstrassproducts oforderm= κ associatedtoset
f f ⌊ ⌋
of zeros and set of poles of f, respectively. Then, a straightforward computation
shows that for z X
f
∈
( 1)m
(m+1,z)= − (logWN(z))(m+1) (logWP(z))(m+1) ,
Zf m! f − f
(cid:16) (cid:17)
and, moreover
(logf(z))(m+1) =(logWN(z))(m+1) (logWP(z))(m+1).
f − f
We have the following theorem.
Theorem 4. Let f be a meromorphic function of zeta-type, of finite order κ. Fix
z X . The superzeta function (s,z) defined by (4.2) has a holomorphic con-
f f
tin∈uation to all s C, through theZequation
∈
sinπs ∞ f′
(4.3) (s,z)= (z+y) y−sdy.
f
Z π f
Z0 (cid:18) (cid:19)
Proof. For z X and κ < Re(s) < 3, we apply Equation 2.3 to (s,z) and
f f,n
∈ Z
(s,z) and proceed analogously as in the proof of (2.7) to get
f,p
Z
(s,z)= N(s,z) P(s,z)
Zf Zf −Zf
( 1)m ∞
= − (logWN(z+y))(m+1) (logWP(z+y))(m+1) ym−sdy
Γ(m+1 s)Γ(s) f − f
− Z0( 1(cid:16))m ∞ (cid:17)
(4.4) = − (logf(z+y))(m+1)ym−sdy.
Γ(m+1 s)Γ(s)
− Z0
For a fixed z X , generalized Dirichlet series representation (4.1) yields the
f
∈
asymptotic behavior
1
(logF (z+y))(l) =O , as y , for all positive integers n
yn →∞
(cid:18) (cid:19)
10 JOSHUAS.FRIEDMAN,JAYJORGENSON,ANDLEJLASMAJLOVIC´
and
(logF (z+y))(l) =O(1), as y 0
ց
forany 1 l m. Therefore,for m 1<Re(s)<m we may integrateby partsin
≤ ≤ −
(4.4) and obtain
( 1)m ∞
(4.5) (s,z)= − ym−sd (logf(z+y))(m) =
f
Z Γ(m s)(m s)Γ(s)
− − Z0 (cid:16) (cid:17)
( 1)m−1 ∞
− (logf(z+y))(m)ym−1−sdy
Γ(m s)Γ(s)
− Z0
Integrating by parts in (4.5) for m 2<Re(s) <m 1 (in case when m 0) we
− − ≥
obtain
( 1)m−2 ∞
(s,z)= − (logf(z+y))(m−1)ym−2−sdy.
f
Z Γ(m 1 s)Γ(s)
− − Z0
Proceeding inductively in m, we deduce that
(4.6)
1 ∞ sinπs ∞ f′
(s,z)= (logf(z+y))′y−sdy = (z+y) y−sdy,
f
Z Γ(1 s)Γ(s) π f
− Z0 Z0 (cid:18) (cid:19)
for 0<Re(s)<1.
First, we claim that for a fixed z X , the integral
f
∈
∞ Z′
I(s,z)= (z+y) y−sdy
Z
Z0 (cid:18) (cid:19)
which appears on the right hand side of (4.6) is actually holomorphic function in
the half plane Re(s) < 1. To see this, let µ 0 be arbitrary. Decay properties of
≤
(logf(z+y))′,asy + withn> µ+2,yieldthat(logf(z+y))′y−s =O(y−2),
asy + ,foralls→such∞thatµ<Re−(s) 0. Moreover,thebound f′(z+y) =O(1),
→ ∞ ≤ f(z+y)
for fixed z X implies that (logf(z+y))′y−s = O(1), as y 0, for all s in the
f
∈ →
half plane Re(s) 0. This shows that for z X the integral I(s,z) is absolutely
f
≤ ∈
convergentin the stripµ<Re(s) 0, hence representsa holomorphic function for
≤
all s in that strip. Since µ 0 was arbitrarily chosen, we have proved that I(s,z),
≤
for z X , is holomorphic function in the half plane Re(s) 0.
f
∈ ≤
Next,weclaimthatI(s,z),forz X ,canbemeromorphicallycontinuedtothe
f
∈
half-plane Re(s)>0 with simple poles at the points s=1,2,... and corresponding
residues
1
(4.7) Res I(s,z)= (logf(z))(n).
s=n
−(n 1)!
−
Since the function sin(πs) has simple zeros at points s = 1,2,... this would prove
that (s,z), for z X is actually anentire function ofs and the proofwouldbe
f +
Z ∈
complete.
Let µ > 0 be arbitrary, put n = µ to be the integer part of µ and let δ > 0
⌊ ⌋
(depending upon z X and µ) be such that for y (0,δ) we have the Taylor
f
∈ ∈
series expansion
n (logf(z))(j)
(logf(z+y))′ = yj−1+R (z,y),
1
(j 1)!
j=1 −
X