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

R. Macaitienė 85
6. Proof of Theorem 2
Let Ω 1 be a subset of Ω such that for ω 1 ∈ Ω 1 the series
∞∑
m=1
ω 1 (m)
(m + α) s
converges uniformly on compact subsets of D and, let for σ > 1 2
, the estimate
be valid. By Lemma 5 we have that m H (Ω 1 ) = 1.
Lemma 9. Let K be a compact subsed of D. Then
for ω 1 ∈ Ω 1 .
lim
lim
n→∞ T →∞
1
T
∫ T
0
∫ T
0
|ζ(σ + it, ω 1 , α)| 2 dt = BT (14)
sup |ζ(s + iτ, ω 1 , α) − ζ n (s + iτ, ω 1 , α)|dτ = 0,
s∈K
Proof. The lemma is proved similarly to Lemma 7, since by (14) and the Cauchy
inequality the estimate
∫ T
follows.
Let
0
|ζ(σ + it, ω 1 , α)|dt = BT
Q T (A) = ν τ T (ζ(s + iτ, ω 1 , α) ∈ A),
A ∈ B(H(D)).
Lemma 10. There exists a probability measure P 1 on (H(D), B(H(D))) such that
the measures P T and Q T both converge weakly to P 1 as T → ∞.
Proof. By Lemma 8 both the measures PT,n 1 and Q1 T,n converge weakly to the
same measure Pn 1 as T → ∞. First we will prove that the family of probability
measures {Pn} 1 is tight.
Let θ be a random variable, uniformly distributed in the interval [0, 1] and
Then by Lemma 8 we have that
X T,n (s, α) = ζ 2,n (s + iT θ, α).
X T,n
D
−→ Xn , T → ∞, (15)