8 2. AN ABSTRACT MEROMORPHICITY THEOREM
for any ξ ΣA
+
and s C. Consider the zeta function
Z(Θ,)(s)
= exp



k=1
1
k
σk
A
(ξ)=ξ
eu(Θ,)(ξ,s)⎠
k

.
One of the main tools used in the proof of Theorem 1.1 is the following gener-
alization of Theorem 1 in [I5].
Theorem 2.1. Let θ (0, 1) and ∆0 0 be constants and let (f, ω)
C(c0, C0, Ω) satisfy the conditions (2.4) and (2.5). Then there exist constants
µ0 = µ0(c0, C0, Ω, ∆0) 0,
0
= 0(µ0, c0, C0, Ω, ∆0) 0 and s0 = s0(f, ω) R
such that for any (0, 0) if
ˆ
f, ˆ ω, Fθ(ΣA)
+
satisfy the conditions:
(i)
ˆ−
f f
θ
C0 ∆0 and ˆ ω ω
θ
∆0 ,
(ii) ∆(ξ) R for any ξ ΣA,
+
and ∆(ξ) = ∆(ξ0, ξ1) (i.e. ∆(ξ) depends on
the first two coordinates of ξ only),
(iii) ∆(ξ) ∆0 for any ξ ΣA
+
with B(ξ0, ξ1) = 0 and |∆(ξ)| C0 ∆0 for
any ξ ΣA
+
with B(ξ0, ξ1) = 1,
then for
ˆ
Θ = (
ˆ
f, ˆ ω, ∆) the following hold:
(a) The zeta function Z(
ˆ
Θ ,)(s) is meromorphic in
Vµ0 = {s C : Re(s) s0 µ0}
and has a pole s with |s s0| µ0. Moreover,
Z(
ˆ
Θ ,)(s)
is analytic for
Re(s) s0.
(b) The pole s can be chosen in such a way that
(2.6) |s s0| C1
∆0/2p
for some constant C1 = C1(p, c0, C0, Ω, ∆0) 0.
Explicit estimates of the constants µ0, 0, C1 and C2 are given in Ch. 6.
Theorem 1 in [I5] deals with the case when just one fixed triple (f, ω, ∆) is
considered. The proof of the above theorem given in Chapters 4-6 below is based
on a further development of Ikawa’s method in [I4], [I5], and is considerably more
difficult.
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