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

R. Macaitienė 81
m(A ′ ) = 0 or m(A ′ ) = 1. Hence it follows that Q(A) = 0 or Q(A) = 1, i. e. the
process |ζ(σ + iτ, ω, α)| 2 is ergodic.
Observing that ˆζ(σ, ϕ τ (ω), α) 0 and using (4), (5) we deduce from Lemma 4
that
∫
1
T
∫
1
T
lim ˆζ(σ, ϕ τ (ω), α)dt = lim |ζ(σ + iτ, ω, α)| 2 dτ = E
T →∞ T
T →∞ T
ˆζ(σ, ω, α)
0
0
for almost all ω ∈ Ω. From this by (4) we obtain the assertion of the lemma.
4. Approximation by mean of the function ζ(s, α)
Let σ 1 > 1 2
. We define the function
l n (s, α) = s σ 1
Γ ( s
σ 1
)
(n + α) s , n ∈ N,
in the strip 1 2 < σ < 1. Moreover, let for σ 1 > 1 2
ζ 2,n (s, α) = 1
2πi
∫
σ 1+i∞
σ 1−i∞
ζ(s + z, α)l n (z, α) dz
z .
Since Γ(σ + it) = Be −c 1|t| , uniformly in σ, |σ| c 1 , the integral for ζ 2,n (s, α) exists.
Lemma 6. Let a and b be positive numbers. Then the formula
is valid.
Proof can be found in [4].
1
2πi
∫
b+i∞
b−i∞
Γ(s)a −s ds = e −a
Lemma 7. Let K be a compact subset of the strip σ > 1 2 . Then
lim
lim
∫ T
n→∞ T →∞
0
sup |ζ(σ + iτ, α) − ζ 2,n (s + iτ, α)|dτ = 0.
s∈K