3 Holomorphicity in s
This section is devoted to extend theorem 2.13 to s ∈ C
p
. We do this by showing that all functions appearing in theorem 2.13 are holomorphic functions in
x, s for kxk
1 and then prove point-wise convergence of these functions. We do this since we need the holomorphicity in the direct proof
of theorem 3.1 see section 3.6.2 and since there are only minor changes between
s fix and s as
variables. We do not introduce here holomorphic functions in more than one variable since we do not need it
in the calculations except in section 3.6.2. A good introduction to holomorphic functions in more than one variable is the book “From holomorphic functions to complex manifolds” [7].
We now state the main theorem of this section
Theorem 3.1. We have
E
Z
s
n
x
→
Y
k∈N
p
\{0}
1 −
x
k
−
s k
−1
k
for n → ∞ and all x, s ∈ C
p
with k xk
1. 3.1
We use the principal branch of logarithm to define a
b
for a ∈ R
≤0
.
3.1 Corollaries of theorem 3.1
Before we prove theorem 3.1, we give some corollaries
Corollary 3.1.1. We have for s
1
, s
2
, x
1
, x
2
∈ C
p
with k x
1
k 1, kx
2
k 1 E
Z
s
1
n
x
1
Z
s
2
n
x
2
→
Y
k
1
, k
2
∈N
p
k
1
+k
2
6=0
1 − x
k
1
1
x
k
2
2 −
s1 k1
s2+k2−1 k2
−1
k1
n → ∞.
Proof. We use the definition of
s k
in 3.2 for s ∈ C, k ∈ N see later. We apply theorem 3.1 for p
′
:= 2p and the identity
−s k
= −1
k s+k−1 k
.
Corollary 3.1.2. We have for x
1
, x
2
, x
3
, x
4
, s
1
, s
2
, s
3
, s
4
∈ C with max |x
i
| 1
E
Z
s
1
n
x
1
Z
s
2
n
x
2
Z
s
3
n
x
3
Z
s
4
n
x
4
→
Y
k
1
,k
2
,k
3
,k
4
∈N k
1
+k
2
+k
3
+k
4
6=0
1 − x
k
1
1
x
k
2
2
x
k
3
3
x
k
4
4
s1 k1
s2 k2
s3+k3−1 k3
s4+k3−1 k4
−1
k1+k2+1
We can also calculate the limit of the Mellin-Fourier-transformation of Z
n
x, as Keating and Snaith did in their paper [11] for the unitary group.
Corollary 3.1.3. We have for s
1
, s
2
∈ R, x ∈ C with |x| 1 E
|Z
n
x|
s
1
e
is
2
argZ
n
x
→
∞
Y
k
1
,k
2
∈N k
1
+k
2
6=0
1 − x
k
1
x
k
2
s1−s2 2
k1 s1+s2
2 k2
−1
k1+1
.
Proof. We have |z|
s
1
= z
s
1
2
z
s
1
2
and e
is
2
ar gz
=
z
s2
|z|
s2
. 1100
3.2 Easy facts and definitions
We simplify the proof of theorem 3.1 by assuming p = 1. We first rewrite 1 − x
m
as r
m
e
i ϕ
m
with r
m
0 and ϕ
m
∈] − π, π] for all m ∈ N.
Convention: we choose 0 r 1 fixed and prove theorem 3.1 for |x| r.
We restrict x to {|x| r} because some inequalities in the next lemma are only true for r 1.
Lemma 3.2. The following hold:
1. 1 − r
m
≤ r
m
≤ 1 + r
m
and | ϕ
m
| ≤ α
m
, where α
m
is defined in figure 1.
Figure 1: Definition of α
m
2. One can find a β
1
1 such that 0 ≤ α
m
≤ β
1
r
m
. 3. For −r
y r one can find a β
2
= β
2
r 1 with | log1 + y| ≤ β
2
| y|. 4. There exists a
β
3
= β
3
r, such that for all m and 0 ≤ y ≤ r 1 + y
m
≤ 1
1 − y
m
≤ 1 + β
3
y
m
. 5. We have for all s ∈ C
Log 1 − x
m s
≡ sLog 1 − x
m
mod 2 πi
with Log. the principal branch of logarithm. Proof. The proof is straight forward. We therefore give only an overview
1. We have |x
m
| r
m
and thus 1 − x
m
lies inside the circle in figure 1. This proves point 1 2. We have that sin
α
m
= r
m
by definition and sinz ∼ z for z → 0. This proves point 2. 3. We have log1 + y = log |1 + y| + iarg1 + y. This point now follows by takeing a look at
log |1 + y| and arg1 + y separately. 4. Obvious.
1101
5. Obvious.
Lemma 3.3. Let Y
m
be a Poisson distributed random variable with E Y
m
=
1 m
. Then E
y
d Y
m
= exp
y
d
− 1 m
for y, d ≥ 0.
3.3 Extension of the definitions
