a limit theorem for the hurwitz zeta-function in the space of analytic ...
a limit theorem for the hurwitz zeta-function in the space of analytic ...
a limit theorem for the hurwitz zeta-function in the space of analytic ...
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84 A <strong>limit</strong> <strong><strong>the</strong>orem</strong> <strong>for</strong> <strong>the</strong> Hurwitz <strong>zeta</strong>-<strong>function</strong>...<br />
It is known that<br />
∫ T<br />
Now it follows from (11) and (12) that<br />
1<br />
T sup<br />
∫2T<br />
σ,s∈L<br />
0<br />
= B sup<br />
0<br />
|ζ(σ 2 + it, α)| 2 dt = BT. (12)<br />
|ζ(σ + it, α) − ζ 2,n (σ + it, α)|dt<br />
∫ ∞<br />
σ,s∈L<br />
−∞<br />
+ B T sup<br />
∫2T<br />
σ,s∈L<br />
0<br />
= B sup<br />
σ∈[− 1 2 + 3ε<br />
|l n (σ 2 − σ + iτ, α)| ( 2 + 2|τ| )<br />
dτ<br />
T<br />
|l n (1 − σ − it, α)|<br />
dt<br />
|1 − σ − it|<br />
∫<br />
2 ,− ε 4 ] ∞<br />
−∞<br />
|l n (σ + it, α)|(1 + |τ|)dτ + o(1)<br />
as T → ∞. From <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> <strong>function</strong> l n (s, α) we deduce that<br />
∫<br />
lim sup |l n (σ + it, α)|(1 + |τ|)dτ = 0.<br />
∞<br />
n→∞<br />
σ∈[− 1 2 + 3ε<br />
2 ,− ε 4 ] −∞<br />
This equality toge<strong>the</strong>r with (13) ir (10) gives <strong>the</strong> assertion <strong>of</strong> <strong>the</strong> <strong><strong>the</strong>orem</strong>.<br />
(13)<br />
5. Limit <strong><strong>the</strong>orem</strong> <strong>in</strong> <strong>the</strong> <strong>space</strong> <strong>of</strong> <strong>analytic</strong> <strong>function</strong>s <strong>for</strong> <strong>the</strong> absolutely<br />
convergent series<br />
Let <strong>for</strong> ω ∈ Ω,<br />
ζ 2,n (s, ω, α) =<br />
∞∑<br />
m=1<br />
{<br />
ω(m)<br />
(m + α) s exp −<br />
( ) σ1<br />
}<br />
(m + α)<br />
.<br />
(n + α)<br />
S<strong>in</strong>ce |ω(m)| = 1, <strong>the</strong> series <strong>for</strong> ζ 2,n (s, ω, α) converges absolutely on <strong>the</strong> half-plane<br />
σ > 1 2<br />
. We def<strong>in</strong>e two probability measures on (H(D), B(H(D)))<br />
P 1 T,n(A) = ν τ T (ζ 2,n (s + iτ, α) ∈ A),<br />
Q 1 T,n(A) = ν τ T (ζ 2,n (s + iτ, ω, α) ∈ A).<br />
Lemma 8. There exists a probability measure P 1 n on (H(D), B(H(D))) such that<br />
both measures P 1 T,n and Q1 T,n converge weakly to P 1 n as T → ∞.<br />
Pro<strong>of</strong> is similar to that <strong>for</strong> <strong>the</strong> case <strong>of</strong> <strong>the</strong> Riemann <strong>zeta</strong>-<strong>function</strong> and uses<br />
Lemma 2.