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

R. Macaitienė 83
1
Let L be a simple closed contour lying in {s ∈ C :
2
< σ < 1} and enclosing
the set K, and let δ denote the distance of L from the set K. Then by the Cauchy
formula we have
sup |ζ(s + iτ, α) − ζ 2,n (s + iτ, α)| 1 ∫
|ζ(z + iτ, α) − ζ 2,n (z + iτ, α)||dz|.
s∈K
2πδ
Therefore we obtain that, for sufficiently large T ,
∫
1
T
sup |ζ(s + iτ, α) − ζ 2,n (s + iτ, α)|dτ
T s∈K
0
= B T δ
∫
L
= B|L|
T δ
∫
|dz|
2T
0
+ B|L|
T δ
L
|ζ(Rez + iτ, α) − ζ 2,n (s + iτ, α)|dτ + B|L|
T δ
sup
∫2T
σ,s∈L
0
The contour L can be chosen so that the inequalities
|ζ(s + it, α) − ζ 2,n (s + it, α)|dt.
σ 1 2 + 3ε
4 , δ ε 4
should hold. Then by (9) we have that for such σ
ζ(s + it, α) − ζ 2,n (s + it, α) = B
∫ ∞
−∞
|ζ(σ 2 + it + iτ, α)||l n (σ 2 − σ + iτ, α)|dτ
+ B|l n(1 − σ − it, α)|
.
|1 − σ − it|
Hence, for the same σ, in virtue of the properties of l n (s, α) and ζ(s, α), we find
that
1
T
∫2T
0
= B
|ζ(σ + it, α) − ζ 2,n (σ + it, α)|dt
∫ ∞
−∞
+ B T
∫2T
0
|l n (σ 2 − σ + iτ, α)| 1 T
|l n (1 − σ − it, α)|
dt.
|1 − σ − it|
|τ|+2T
∫
−|τ|
|ζ(σ 2 + it, α)|dtdτ
(11)