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

≤ Ω ∆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.