a limit theorem for the hurwitz zeta-function in the space of analytic ...

80 A limit theorem for the Hurwitz zeta-function...
Lemma 5. Let T → ∞ and 1 2
< σ < 1. Then
∫ T
0
|ζ(σ + it, ω, α)| 2 dt = BT
for almost all ω ∈ Ω.
Proof. Let
Then
Denote
ζ m (σ, ω, α) =
ζ(σ, ω, α) =
ω(m)
(m + α) σ , m ∈ N 0.
∞∑
ζ m (σ, ω, α).
m=0
ˆζ(σ, ω, α) = |ζ(σ, ω, α)| 2 .
Since
E|ζ m (σ, ω, α)| 2 =
1
(m + α) 2σ ,
and the random variables ζ m (σ, ω, α) are pairwise ortogonal, we find that
It is obvious that
E|ˆζ(σ, ω, α)| =
∞∑
E|ζ m (σ, ω, α)| 2 =
m=0
∞∑
m=1
1
< ∞. (4)
(m + α)
2σ
ˆζ(σ, ϕ τ (ω), α) = |ζ(σ, a τ ω, α)| 2 = |ζ(σ + iτ, ω, α)| 2 . (5)
Since the Haar measure is invariant, the equality m H (ϕ τ (A)) = m H (A) is valid
for each A ∈ B(Ω) and every τ ∈ R. Therefore |ζ(σ + iτ, ω, α)| 2 is a strongly
stationary process. It is also an ergodic process. In fact, let A be an invariant set
of |ζ(σ + iτ, ω, α)| 2 , i. e.
Q(A∆A u ) = 0. (6)
We have that
A ′
A ′ u
def
= {ω ∈ Ω : |ζ(σ + iτ, ω, α)| 2 ∈ A} = {ω ∈ Ω : |ζ(σ, a τ ω, α)| 2 ∈ A},
def
= {ω ∈ Ω : |ζ(σ + iτ, ω, α)| 2 ∈ A u } = {ω ∈ Ω : |ζ(σ + iτ + iu, ω, α)| 2 ∈ A}
= {ω ∈ Ω : |ζ(σ + iτ, a u ω, α)| 2 ∈ A}.
Therefore A ′ u = ϕ u (A ′ ) and (A∆A u ) ′ = A ′ ∆A ′ u. From this and from (6) we deduce
that m H (A ′ ∆A ′ u) = m H ((A∆A u ) ′ ) = Q(A∆A u ), that is A ′ is an invariant set with
respect to ϕ τ . But, by Lemma 3, the group {ϕ τ : τ ∈ R} is ergodic. Therefore