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

R. Macaitienė 77
Denote by γ the unit circle on C, i. e. γ = {s ∈ C : |s| = 1}, and let
Ω = ∏ m
γ m ,
where γ m = γ for each m = 0, 1, . . .. With the product topology and pointwise multiplication
Ω is a compact Abelian topological group. Then there exists the probability
Haar measure m H on (Ω, B(Ω)). This yields a probability space (Ω, B(Ω), m H ).
Denote by ω(m) the projection of ω ∈ Ω to the coordinate space γ m .
Let ζ(s, ω, α) be the H(D)-valued random element on (Ω, B(Ω), m H ) given by
the formula
∞∑ ω(m)
ζ(s, ω, α) =
(m + α) s .
m=1
Denote by P ζ the distribution of the random element ζ(s, ω, α), i. e.
P ζ (A) = m H (ω ∈ Ω : ζ(s, ω, α) ∈ A),
A ∈ B(H(D)).
The aim of this note is to prove the following statement. Suppose α is a transcendental
number.
Theorem 2. The probability measure P T converges weakly to P ζ as T → ∞.
2. Auxiliary results
Let
p n (s, α) =
n∑
m=1
1
(m + α) s
be an arbitrary Dirichlet polynomial, D denote some open subset of C, and let α is
a transcendental number. We define the probability measure
P T,pn (A) = ν τ T (p n (s + iτ, α) ∈ A),
A ∈ B(H(D)).
Lemma 1. There exists a probability measure P pn on (H(D), B(H(D))) such that
P T,pn converges weakly to P pn as T → ∞.
Proof. Let
Ω m =
n∏
γ m , γ m = γ.
m=0
Let us define the function h : Ω n → H(D) by the formula
h(x 0 , . . . , x n ) =
n∑
m=0
1
(m + α) s 1
x m
, (x 0 , . . . , x n ) ∈ Ω n .