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

78 A limit theorem for the Hurwitz zeta-function...
The function h is continuous on Ω n , and
p n (s + iτ, α) = h(α iτ , (1 + α) iτ , . . . , (n + α) iτ ). (1)
Now we define the probability measure
Q T (A) = νT τ ((α iτ , (1 + α) iτ , . . . , (n + α) iτ ) ∈ A)
on (Ω n , B(Ω n )). The Fourier transform g T (k 0 , . . . , k n ), k j ∈ Z, j = 1, . . . , n, of Q T
is given by the formula
∫
g T (k 0 , . . . , k n ) =
Ω
x k 0
0 , . . . , xkn n dQ T = 1 T
∫ T
∫ T
0
n∏
(j + α) ikmτ dτ
j=0
= 1 exp ( n∑
iτ k m log(m + α) ) dτ
T
0
m=0
⎧
1 if (k 0 , . . . , k n ) = (0, . . . , 0),
⎪⎨
( n∑
)
exp iT k m log(m+α)
=
m=0
n∑
if (k 0 , . . . , k n ) ≠ (0, . . . , 0).
⎪⎩ iT k m log(m+α)
m=0
(2)
Since α is transcendental, log(m + α), m = 0, 1, . . . , n, are linearly independent
over the field of rational numbers, whence we find that
g T (k 0 , . . . , k n ) =
{
1 if (k0 , . . . , k n ) = (0, . . . , 0),
0 if (k 0 , . . . , k n ) ≠ (0, . . . , 0)
as T → ∞. Therefore, the measure Q T converges weakly to the Haar measure m nH
on (Ω, B(Ω)) as T → ∞. Taking into account the continuity of the function h and
the formula (1), and applying Theorem 5.1 from [1], we obtain that the probability
measure P T,pn converges weakly to the measure m nH h −1 as T → ∞. The theorem
is proved.
Let g(m), m ∈ N, be an arithmetic function, |g(m)| = 1,
p n (s, g, α) =
n∑
m=1
g(m)
(m + α) s ,
and
˜P T,pn (A) = ν τ T
(
pn (s + iτ, g, α) ∈ A ) , A ∈ B(H(D)).
Lemma 2. The probability measures P T,pn and ˜P T,pn both converge weakly to the
same measure as T → ∞.