Discrete limit theorems for Estermann zeta-functions. II
A discrete limit theorem in the sense of weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function with explicitly given the limit measure is proved.
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Discrete limit theorems for Estermann zeta-functions. II / A. Laurincikas, R. Macaitiene // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 69–83. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1533652019-06-15T01:25:38Z Discrete limit theorems for Estermann zeta-functions. II Laurincikas, A. Macaitiene, R. A discrete limit theorem in the sense of weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function with explicitly given the limit measure is proved. 2008 Article Discrete limit theorems for Estermann zeta-functions. II / A. Laurincikas, R. Macaitiene // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 69–83. — Бібліогр.: 9 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 11M41. http://dspace.nbuv.gov.ua/handle/123456789/153365 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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A discrete limit theorem in the sense of weak convergence of probability measures in the space of meromorphic functions for the Estermann zeta-function with explicitly given the limit measure is proved. |
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Article |
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Laurincikas, A. Macaitiene, R. |
| spellingShingle |
Laurincikas, A. Macaitiene, R. Discrete limit theorems for Estermann zeta-functions. II Algebra and Discrete Mathematics |
| author_facet |
Laurincikas, A. Macaitiene, R. |
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Laurincikas, A. |
| title |
Discrete limit theorems for Estermann zeta-functions. II |
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Discrete limit theorems for Estermann zeta-functions. II |
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Discrete limit theorems for Estermann zeta-functions. II |
| title_fullStr |
Discrete limit theorems for Estermann zeta-functions. II |
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Discrete limit theorems for Estermann zeta-functions. II |
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discrete limit theorems for estermann zeta-functions. ii |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/153365 |
| citation_txt |
Discrete limit theorems for Estermann zeta-functions. II / A. Laurincikas, R. Macaitiene // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 3. — С. 69–83. — Бібліогр.: 9 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT laurincikasa discretelimittheoremsforestermannzetafunctionsii AT macaitiener discretelimittheoremsforestermannzetafunctionsii |
| first_indexed |
2025-07-14T04:35:35Z |
| last_indexed |
2025-07-14T04:35:35Z |
| _version_ |
1837595608511152128 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2008). pp. 69 – 83
c© Journal “Algebra and Discrete Mathematics”
Discrete limit theorems for Estermann
zeta-functions. II
Antanas Laurinčikas, Renata Macaitienė
Communicated by V. V. Kirichenko
Abstract. A discrete limit theorem in the sense of weak
convergence of probability measures in the space of meromorphic
functions for the Estermann zeta-function with explicitly given the
limit measure is proved.
1. Introduction
Let s = σ + it be a complex variable, k and l be coprime integers, and,
for α ∈ C,
σα(m) =
∑
d/m
dα.
For σ > max(1, 1 + ℜα), the Estermann zeta-function E(s; k
l , α) with
parameters k
l and α is defined by
E
(
s;
k
l
, α
)
=
∞∑
m=1
σα(m)
ms
exp
{
2πim
k
l
}
.
The function E(s; k
l , α) has analytic continuation to the whole complex
plane, except for two simple poles at s = 1 and s = 1 + α if α 6= 0, and
a double pole at s = 1 if α = 0. In view of the equation
E
(
s;
k
l
, α
)
= E
(
s − α;
k
l
,−α
)
,
Partially supported by Lithuanian Foundation of Studies and Science.
2000 Mathematics Subject Classification: 11M41.
Key words and phrases: Estermann zeta-function, Haar measure, limit theo-
rem, probability measure, weak convergence.
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.70 Discrete limit theorems for Estermann zeta-functions
we may suppose that ℜα ≤ 0.
The present paper is a continuation of [6], where a discrete limit
theorem on the complex plane for E(s; k
l , α) has been proved. To state
the latter theorem, we need some definitions and notation. Denote by
B(S) the class of Borel sets of the space S. Moreover, let
Ω =
∏
p
γp,
where γp = {s ∈ C : |s| = 1}
def
=γ for each prime p. The torus Ω is a
compact topological Abelian group, therefore, on (Ω,B(Ω)) the proba-
bility Haar measure mH can be defined. This gives a probability space
(Ω,B(Ω), mH). Denote by ω(p) the projection of ω ∈ Ω to the coordinate
space γp, p ∈ P (P denotes the set of all prime numbers), and put, for
m ∈ N,
ω(m) =
∑
pα‖m
ωα(p),
where pα ‖ m means that pα | m but pα+1 ∤ m. Now suppose that
ℜα ≤ 0 and on the probability space (Ω,B(Ω), mH) define the complex-
valued random element E(σ; k
l , α; ω), for σ > 1
2
, by
E
(
σ;
k
l
, α; ω
)
=
∞∑
m=1
σα(m)ω(m)
mσ
exp
{
2πim
k
l
}
.
Let P C
E,σ be the distribution of E(σ; k
l , α; ω), i.e.,
P C
E,σ(A) = mH
(
ω ∈ Ω : E
(
σ;
k
l
, α; ω
)
∈ A
)
, A ∈ B(C).
In the sequel, for N ∈ N0 = N
⋃
{0}, we will use the notation
µN (...) =
1
N + 1
∑
0≤m≤N
...
1,
where in place of dots a condition satisfied by m is to written. In [6], the
following statement has been proved.
Theorem 1. Suppose that ℜα ≤ 0 and σ > 1
2
, and that h > 0 is a fixed
number such that exp
{
2πr
h
}
is irrational for all r ∈ Z \ {0}. Then the
probability measure
µN
(
E
(
σ + imh;
k
l
, α
)
∈ A
)
, A ∈ B(C),
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.A. Laurinčikas, R. Macaitienė 71
converges weakly to P C
E,σ as N → ∞.
The function E(s; k
l , α) is meromorphic one. Therefore, its asymptotic
behavior is better reflected by a limit theorem in the space of meromor-
phic functions.
Let C∞ = C∪{∞} be the Riemann sphere with the metric d defined
by
d(s1, s2) =
2|s1 − s2|√
1 + |s1|2
√
1 + |s2|2
, d(s,∞) =
2√
1 + |s|2
, d(∞,∞) = 0,
s, s1, s2 ∈ C. Let G be a region on the complex plane. Denote by M(G)
the space of meromorphic on G functions f : G → (C∞, d) equipped
with the topology of uniform convergence on compacta. In this topology,
a sequence {fn} ⊂ M(G) converges to f ∈ M(G) if, for every compact
subset K ⊂ G,
lim
n→∞
sup
s∈K
d(fn(s), f(s)) = 0.
All analytic functions on G form a subspace H(G) of M(G).
Let D =
{
s ∈ C : σ > 1
2
}
. Then, in the case ℜα ≤ 0,
E
(
s;
k
l
, α; ω
)
=
∞∑
m=1
σα(m)ω(m)
ms
exp
{
2πim
k
l
}
,
is an H(D)-valued random element defined on the probability space
(Ω,B(Ω), mH). Denote by PH
E its distribution given, for A ∈ B(H(D)),
by
PH
E (A) = mH
(
ω ∈ Ω : E
(
s;
k
l
, α; ω
)
∈ A
)
,
and define the probability measure
PN (A) = µN
(
E
(
s + imh;
k
l
, α
)
∈ A
)
, A ∈ B(M(D)).
The aim of this paper is to prove a limit theorem for the measure PN .
Theorem 2. Suppose that ℜα ≤ 0 and that h > 0 is a fixed number
such that exp
{
2πr
h
}
is irrational for all r ∈ Z \ {0}. Then the probability
measure PN converges weakly to PH
E as N → ∞.
We suppose in the sequel that ℜα ≤ 0, and that exp
{
2πr
h
}
is irrational
for all r ∈ Z \ {0}.
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2. Case of absolute convergence
In this section, we will prove a discrete limit theorem in the space of
analytic functions for a function given by absolutely convergent Dirichlet
series and related to the function E
(
s; k
l , α
)
.
Let, for brevity, s1 = 1, s2 =
{
1 + α if α 6= 0,
1 if α = 0,
and
f(s) =
2∏
j=1
(
1 − 2sj−s
)
.
Then f(sj) = 0, j = 1, 2, and the point s = 1 is a double zero of f(s) if
α = 0. Define
Ê
(
s;
k
l
, α
)
= f(s)E
(
s;
k
l
, α
)
Then, clearly, Ê
(
s; k
l , α
)
is an analytic function on the half-plane D.
Moreover, denoting by |A| the number of elements of a set A, we have
that, for σ > 1,
Ê
(
s;
k
l
, α
)
=
2∏
j=1
(
1 −
2sj
2s
) ∞∑
m=1
σα(m)
ms
exp
{
2πim
k
l
}
=
∑
A⊆{1,2}
∞∑
m=1
σα(m)exp
{
2πim
k
l
}
2
∑
j∈A
sj
(−1)|A|2−|A|sm−s
=
2∑
j=0
∞∑
m=1
am,j
(
k
l
, α
)
1
2jsms
.
It is easily seen that, for all m ∈ N and j = 0, 1, 2,
am,j
(
k
l
, α
)
≪ |σα(m)|.
Let σ1 > 1
2
be a fixed number, and, for m, n ∈ N,
vn(m) = exp
{
−
(m
n
)σ1
}
.
Define
Ên
(
s;
k
l
, α
)
=
2∑
j=0
∞∑
m=1
am,j
(
k
l , α
)
vn(m)
2jsms
,
and, for ω̂ ∈ Ω,
Ên
(
s;
k
l
, α; ω̂
)
=
2∑
j=0
∞∑
m=1
am,j
(
k
l , α
)
ω̂j(2)ω̂(m)vn(m)
2jsms
.
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.A. Laurinčikas, R. Macaitienė 73
It was observed in [5] that the above series both converge absolutely for
σ > 1
2
. This section is devoted to the weak convergence of probability
measures
PN,n = µN
(
Ên
(
s + imh;
k
l
, α
)
∈ A
)
, A ∈ B(H(D)),
and
P̂N,n = µN
(
Ên
(
s + imh;
k
l
, α; ω̂
)
∈ A
)
, A ∈ B(H(D)).
Theorem 3. There exists a probability measure Pn on (H(D),B(H(D)))
such that both the measures PN,n and P̂N,n converge weakly to Pn as
N → ∞.
The proof of Theorem 3 is based on a discrete limit theorem on the
torus Ω. Define
QN (A) = µN
(
(p−imh : p ∈ P) ∈ A
)
, A ∈ B(Ω).
Lemma 4. The probability measure QN converges weakly to the Haar
measure mH on (Ω,B(Ω)) as N → ∞.
Proof of the lemma is given in [6], Lemma 5.
Proof of Theorem 3. Define the function un : Ω → H(D) by the
formula
un(ω) =
2∑
j=0
∞∑
m=1
am,j
(
k
l , α
)
vn(m)ωj(2)ω(m)
2jsms
.
From the absolute convergence for σ > 1
2
of the series Ê(s; k
l , α), we have
that the function un is continuous. Moreover, the equality
un
(
(p−imh : p ∈ P)
)
= Ên
(
σ + imh;
k
l
, α
)
holds. Thus, PN,n, = QNu−1
n . This, the continuity of un, Lemma 4
and Theorem 5.1 of [1] show that the measure PN,n converges weakly to
mHu−1
n as N → ∞.
Similarly, in the case of the measure P̂N,n, we define the function
ûn : Ω → H(D) by the formula
ûn(ω) =
2∑
j=0
∞∑
m=1
am,j
(
k
l , α
)
ω̂j(2)ω̂(m)ωj(2)ω(m)vn(m)
2jsms
.
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.74 Discrete limit theorems for Estermann zeta-functions
Then in the above way we obtain that the measure P̂N,n converges weakly
to mH û−1
n as N → ∞. So, it remains to prove that the measures mHu−1
n
and mH û−1
n coincide. Let, for ω ∈ Ω, u(ω) = ωω̂. Then
ûn(ω) = un(ωω̂) = un(u(ω)).
Therefore, using the invariance of the Haar measure mH , we find that
mH û−1
n = mH(un(u))−1 = (mHu−1)u−1
n = mHu−1
n ,
and the theorem is proved.
We note that the requirement on the irrationality of exp
{
2πr
h
}
, r ∈
Z \ {0}, is used in the proof of Lemma 4, hence also for the proof of
Theorem 3.
3. Approximation results
Let, for ω ∈ Ω and s ∈ D,
Ê
(
s;
k
l
, α; ω
)
=
2∑
j=0
∞∑
m=1
am,j
(
k
l
, α
)
ωj(2)ω(m)
2jsms
=
2∏
j=1
(
1 −
2sjω(2)
2s
) ∞∑
m=1
σα(m)ω(m)
ms
exp
{
2πim
k
l
}
.
Then Ê
(
s; k
l , α; ω
)
is an H(D)-valued random element defined on the
probability space (Ω,B(Ω), mH). Denote by P
Ê
the distribution of
Ê
(
s; k
l , α; ω
)
. In this section, we approximate in the mean the functions
Ê
(
s; k
l , α
)
and Ê
(
s; k
l , α; ω
)
by Ên
(
s; k
l , α
)
and Ên
(
s; k
l , α; ω
)
, respec-
tively.
Theorem 5. Let K be a compact subset of D. Then
lim
n→∞
lim sup
N→∞
1
N + 1
N∑
m=0
sup
s∈K
∣∣∣∣Ê
(
s + imh;
k
l
, α
)
− Ên
(
s + imh;
k
l
, α
)∣∣∣∣ = 0.
Proof. For n ∈ N, define
ln(s) =
s
σ1
Γ
(
s
σ1
)
ns,
where Γ(s) is the Euler gamma function and σ1 is defined in Section 2.
Then, see, [5], for σ > 1
2
,
Ên
(
s;
k
l
, α
)
=
1
2πi
σ1+i∞∫
σ1−i∞
Ê
(
s + z;
k
l
, α
)
ln(z)
dz
z
. (1)
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.A. Laurinčikas, R. Macaitienė 75
Suppose that min{σ : s ∈ K} = 1
2
+ η, η > 0. Now we take σ2 = 1
2
+ η
2
and using (1) obtain by the residue theorem that, for σ > σ2,
Ên
(
s;
k
l
, α
)
=
1
2πi
σ2−σ+i∞∫
σ2−σ−i∞
Ê
(
s + z;
k
l
, α
)
ln(z)
dz
z
+ Ê
(
s;
k
l
, α
)
.
(2)
Let L be a simple closed contour lying in D and enclosing the set K, and
let δ be the distance of L from K. The an application of the Cauchy
integral formula yields the estimate
sup
s∈K
∣∣∣∣Ê
(
s + imh;
k
l
, α
)
− Ên
(
s + imh;
k
l
, α
)∣∣∣∣
≤
1
2πδ
∫
L
∣∣∣∣Ê
(
z + imh;
k
l
, α
)
− Ên
(
z + imh;
k
l
, α
)∣∣∣∣ |dz|.
Therefore, taking into account (2), we find that
1
N + 1
N∑
m=0
sup
s∈K
∣∣∣∣Ê
(
s + imh;
k
l
, α
)
− Ên
(
s + imh;
k
l
, α
)∣∣∣∣
≪
|L|
Nδ
sup
σ+iu∈L
N∑
m=0
∣∣∣∣Ê
(
σ + imh + iu;
k
l
, α
)
− Ên
(
σ + imh + iu;
k
l
, α
)∣∣∣∣
≪ sup
σ+iu∈L
∞∫
−∞
|ln(σ2 − σ + iτ)|
|σ2 − σ + iτ |
(
1
N
N∑
m=0
∣∣∣∣Ê
(
σ2 + iu + iτ + imh;
k
l
, α
)∣∣∣∣
)
dτ
≪ sup
σ+iu∈L
∞∫
−∞
|ln(σ2 − σ + iτ)|
|σ2 − σ + iτ |
(
1
N
N∑
m=0
∣∣∣∣Ê
(
σ2 + iu + iτ + imh;
k
l
, α
)∣∣∣∣
2
) 1
2
dτ.
(3)
Since σ2 > 1
2
and ℜα ≤ 0, we have by [9] that
T∫
0
∣∣∣∣E
(
σ2 + it;
k
l
, α
)∣∣∣∣
2
dt ≪ T.
Hence, it follows that also
T∫
0
∣∣∣∣Ê
(
σ2 + it;
k
l
, α
)∣∣∣∣
2
dt ≪ T, (4)
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.76 Discrete limit theorems for Estermann zeta-functions
and
T∫
0
∣∣∣∣Ê
′
(
σ2 + it;
k
l
, α
)∣∣∣∣
2
dt ≪ T. (5)
We choose the contour L to satisfy δ = η
4
. Then u is bounded, and the
Gallagher lemma, see [8], Lemma 1.4, together with estimates (4) and
(5) shows that
1
N
N∑
m=0
∣∣∣∣Ê
(
σ2 + iu + iτ + imh;
k
l
, α
)∣∣∣∣
2
≪
1
Nh
Nh∫
0
∣∣∣∣Ê
(
σ2 + iu + iτ + it;
k
l
, α
)∣∣∣∣
2
dt
+
1
N
Nh∫
0
∣∣∣∣Ê
′
(
σ2 + iu + iτ + it;
k
l
, α
)∣∣∣∣
2
dt
·
Nh∫
0
∣∣∣∣Ê
(
σ2 + iu + iτ + it;
k
l
, α
)∣∣∣∣
2
dt
1
2
≪
1
N
(N + |τ |) ≪ 1 + |τ |. (6)
This and (3) lead to the estimate
1
N + 1
N∑
m=0
sup
s∈K
∣∣∣∣Ê
(
s + imh;
k
l
, α
)
− Ên
(
s + imh;
k
l
, α
)∣∣∣∣
≪ sup
σ+iu∈L
∞∫
−∞
|ln(σ2 − σ + iτ)|(1 + |τ |)dτ. (7)
By the definition of σ2 and the contour L, we have that σ2 − σ ≤ −η
4
for
σ + iu ∈ L. Moreover, the definition of the function ln(s) shows that, for
σ < 0,
lim
n→∞
∞∫
−∞
|ln(σ + iτ)| (1 + |τ |)dt = 0.
Therefore, this and (7) imply the assertion of the lemma.
Theorem 6. Let K be a compact subset of D. Then, for almost all
ω ∈ Ω,
lim
n→∞
lim sup
N→∞
1
N + 1
N∑
m=0
sup
s∈K
∣∣∣∣Ê
(
s + imh;
k
l
, α; ω
)
− Ên
(
s + imh;
k
l
, α; ω
)∣∣∣∣ = 0.
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Proof. In [5] it was observed that, for σ > 1
2
, the estimate
T∫
0
∣∣∣∣Ê
(
σ + it;
k
l
, α; ω
)∣∣∣∣
2
dt ≪ T
holds for almost all ω ∈ Ω. Therefore, the proof repeats the arguments
used in the proof of Theorem 5.
4. Limit theorems for Ê
(
s; k
l
, α
)
On (H(D),B(H(D))), define two probability measures
QN (A) = µN
(
Ê
(
s + imh;
k
l
, α
)
∈ A
)
,
and, for ω ∈ Ω,
Q̂N (A) = µN
(
Ê
(
s + imh;
k
l
, α; ω
)
∈ A
)
.
Theorem 7. There exists a probability measure Q on (H(D),B(H(D)))
such that both the measures QN and Q̂N converge weakly to Q as N → ∞.
Proof. By Theorem 3, the probability measures PN,n and P̂N,n both
converge weakly to the measure Pn. Let θN be a random variable defined
on a certain probability space (Ω̂,B(Ω̂), P) with the distribution
P(θN = mh) =
1
N + 1
, m = 0, 1, ..., N.
Define
XN,n = XN,n(s) = Ên
(
s + iθN ;
k
l
, α
)
,
and denote by Xn = Xn(s) the H(D)-valued random element with the
distribution Pn. Then Theorem 3 implies the relation
XN,n
D
−→
N→∞
Xn, (8)
where, as usual,
D
−→ denotes the convergence in distribution.
The further proof requires a metric on H(D) which induces its topol-
ogy of uniform convergence on compacta. It is known, see, for example,
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.78 Discrete limit theorems for Estermann zeta-functions
[2], that there exists a sequence {Kn : n ∈ N} of compact subsets of D
such that D =
∞⋃
n=1
Kn, Kn ⊂ Kn+1, and if K is a compact of the region
D, then K ⊆ Kn for some n. Then it is easily seen that
ρ(f, g) =
∞∑
n=1
2−n
sup
s∈Kn
|f(s) − g(s)|
1 + sup
s∈Kn
|f(s) − g(s)|
is the mentioned metric.
For every Mr > 0, the Chebyshev inequality yields
P
(
sup
s∈Kr
|XN,n(s)| > Mr
)
= µN
(
sup
s∈Kr
∣∣∣∣Ên
(
s + imh;
k
l
, α
)∣∣∣∣ > Mr
)
≤
1
Mr(N + 1)
N∑
m=0
sup
s∈Kr
∣∣∣∣Ên
(
s + imh;
k
l
, α
)∣∣∣∣ . (9)
Let Lr be a simple closed contour in D enclosing the set Kr, and let δr
be the distance of Lr from Kr. Then by the Cauchy integral formula
sup
s∈Kr
∣∣∣∣Ê
(
s + imh;
k
l
, α
)∣∣∣∣≪
1
δr
∫
Lr
∣∣∣∣Ê
(
z + imh;
k
l
, α
)∣∣∣∣ |dz|.
Therefore, in view of Theorem 5 and (6),
lim sup
N→∞
1
N + 1
N∑
m=0
sup
s∈Kr
∣∣∣∣Ên
(
s + imh;
k
l
, α
)∣∣∣∣
≤ lim sup
N→∞
1
N + 1
N∑
m=0
sup
s∈Kr
∣∣∣∣Ê
(
s + imh;
k
l
, α
)
− Ên
(
s + imh;
k
l
, α
)∣∣∣∣
+ lim sup
N→∞
1
N + 1
N∑
m=0
sup
s∈Kr
∣∣∣∣Ê
(
s + imh;
k
l
, α
)∣∣∣∣
≤ C1r + lim sup
N→∞
|Lr|
δr(N + 1)
sup
σ+iu∈Lr
N∑
m=0
∣∣∣∣Ê
(
s + iu + imh;
k
l
, α
)∣∣∣∣
≤ C1r + C2r
def
=Cr < ∞. (10)
Now let ǫ > 0 be an arbitrary number. We take Mr = Mr,ǫ = Cr
2r
ǫ .
Then we deduce from (9) and (10) that
lim sup
N→∞
P
(
sup
s∈Kr
|XN,n(s)| > Mr,ǫ
)
<
ǫ
2r
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for all n, r ∈ N. Since (8) implies the relation
sup
s∈Kr
|XN,n(s)|
D
−→
N→∞
sup
s∈Kr
|Xn(s)|,
hence we find that
P
(
sup
s∈Kr
|Xn(s)| > Mr,ǫ
)
<
ǫ
2r
(11)
for all n, r ∈ N. Define
Hǫ = {f ∈ H(D) : sup
s∈Kr
|f(s)| ≤ Mr,ǫ r ≥ 1}.
Then the set Hǫ is compact on H(D), and, by (11),
P(Xn(s) ∈ Hǫ) ≥ 1 − ǫ
for all n ∈ N. This means that the family of probability measures {Pn :
n ∈ N} is tight. Therefore, by the Prokhorov theorem, see, for example,
[1], it is relatively compact. Thus, there exists a subsequence {Pnk
} ⊂
{Pn} such that Pnk
converges weakly to some probability measure Q on
(H(D),B(H(D))) as k → ∞. Then also the relation
Xnk
D
−→
k→∞
Q (12)
holds.
Now let
XN = XN (s) = Ê
(
s + iθN ;
k
l
, α
)
.
Then, by Theorem 5, for every ǫ > 0,
lim
n→∞
lim sup
N→∞
P
(
ρ (XN (s),XN,n(s)) ≥ ǫ
)
= lim
n→∞
lim sup
N→∞
µN
(
ρ
(
Ê
(
s + imh;
k
l
, α
)
, Ên
(
s + imh;
k
l
, α
))
≥ ǫ
)
≤ lim
n→∞
lim sup
N→∞
1
(N + 1)ǫ
N∑
m=0
ρ
(
Ê
(
s + imh;
k
l
, α
)
, Ên
(
s + imh;
k
l
, α
))
= 0.
Since the space H(D) is separable, this, (8), (12) together with Theo-
rem 4.2 of [1] show that
XN
D
−→
N→∞
Q. (13)
This means that the measure QN converges weakly to Q as N → ∞.
Moreover, (13) shows that the measure P is independent of the subse-
quence {Pnk
}. Since {Pn} is relatively compact, hence we deduce that
Xn
D
−→
n→∞
Q. (14)
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Now let
X̂N,n(s) = Ên
(
s + iθN ;
k
l
, α; ω
)
,
and
X̂N (s) = Ên
(
s + iθN ;
k
l
, α; ω
)
.
Then, repeating the above arguments for X̂N,n(s) and X̂N (s), applying
Theorems 3 and 6, as well as taking into account (14), we obtain that
the measure Q̂N also converges weakly to Q as N → ∞. The theorem is
proved.
Theorem 8. The probability measure QN converges weakly to P
Ê
as
N → ∞.
Proof. We start with elements of the ergodic theory. Let ah = {p−ih :
p ∈ P}, and fh(ω) = ahω, ω ∈ Ω. Then fh is a measurable measure
preserving transformation on (Ω,B(Ω), mH). It was obtained in [3] that
this transformation is ergodic.
Let A ∈ B(H(D)) be an arbitrary continuity set of the limit measure
Q in Theorem 7. Then, by the latter theorem,
lim
N→∞
µN
(
Ê
(
s + imh;
k
l
, α
)
∈ A
)
= Q(A). (15)
On the space (Ω,B(Ω), mH), define the random variable θ by the formula
θ = θ(ω) =
{
1 if Ê
(
σ; k
l , α; ω
)
∈ A,
0 if Ê
(
σ; k
l , α; ω
)
/∈ A.
Then, denoting by Eθ the expectation of θ, we have that
Eθ =
∫
Ω
θdmH = mH
(
ω ∈ Ω : Ê
(
s;
k
l
, α; ω
)
∈ A
)
= P
Ê
(A). (16)
Since the transformation fh is ergodic, the classical Birkhoff–Khinchine
theorem, see, for example, [4], shows that
lim
N→∞
1
N + 1
N∑
m=0
θ
(
fm
h (ω)
)
= Eθ (17)
for almost all ω ∈ Ω. On the other hand, from the definitions of fh and
θ, we deduce that
1
N + 1
N∑
m=0
θ
(
fm
h (ω)
)
= µN
(
Ê
(
s + imh;
k
l
, α; ω
)
∈ A
)
.
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This, (16) and (17) give the equality
lim
N→∞
µN
(
Ê
(
s + imh;
k
l
, α; ω
)
∈ A
)
= P
Ê
(A).
Therefore, in view of (15),
Q(A) = P
Ê
(A) (18)
for all continuity sets of the measure Q. Since all continuity sets con-
stitute the determining class, (18) holds for all A ∈ B(H(D)), and the
theorem is proved.
5. Two-dimensional theorem
Let H2(D) = H(D) × H(D), and
f(s, ω) =
2∏
j=1
(
1 −
2sjω(2)
2s
)
.
On the probability space (Ω,B(Ω), mH), define an H2(D)-valued random
element F (s, ω) by
F (s, ω) =
(
f(s, ω), Ê
(
s;
k
l
, α; ω
))
.
In this section, we consider the weak convergence of the probability mea-
sure
RN (A)
def
=µN
((
f(s + imh), Ê
(
s + imh;
k
l
, α
))
∈ A
)
, A ∈ B(H2(D)).
Theorem 9. The probability measure RN converges weakly to the distri-
bution PF of the random element F (s, ω) as N → ∞.
Proof. The function f(s) is a Dirichlet polynomial. Therefore, the
probability measure
µN (f(s + imh) ∈ A) , A ∈ B(H(D)),
converges weakly to the distribution of the random element f(s, ω) as
N → ∞. Now this, Theorem 8 and an application of the modified
Cramér–Wald criterion, an example of its application is given in [7], leads
to the statement of the theorem.
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6. Proof of the main theorem
Theorem 2 is a consequence of Theorem 9.
Proof of Theorem 2. It is not difficult to see that, for the metric d
defined in Section 1, the equality
d(g1, g2) = d
(
1
g1
,
1
g2
)
, g1, g2 ∈ H(D),
holds. Therefore, the function u : H2(D) → M(D) defined by the formula
u(g1, g2) =
g2
g1
, g1, g2 ∈ H(D),
is continuous, and PN = RNu−1. Hence, by Theorem 5.1 of [1] and
Theorem 9, the measure PN converges weakly to the measure PF u−1,
i.e., to
mH
(
ω ∈ Ω :
Ê
(
s; k
l , α; ω
)
f(s, ω)
∈ A
)
, A ∈ B(M(D)). (19)
However, by the definition of the random element Ê
(
s; k
l , α; ω
)
, we have
that
Ê
(
s; k
l , α; ω
)
f(s, ω)
=
∞∑
m=1
σα(m)ω(m)
ms
exp
{
2πim
k
l
}
= E
(
s;
k
l
, α; ω
)
.
Therefore, (19) coincides with
mH
(
ω ∈ Ω : E
(
s;
k
l
, α; ω
)
∈ A
)
, A ∈ B(H(D)).
The theorem is proved.
References
[1] P. Billingsley, Convergence of Probability Measures. Wiley, New York, 1968.
[2] J. B. Conway, Functions of One Complex Variable. Springer – Verlag, New York,
1973.
[3] R. Kačinskaitė, A discrete limit theorem for the Matsumoto zeta-function on the
complex plane. Lith. Math. J., 40(4) (2000), 364–378.
[4] U. Krengel, Ergodic Theorems. Walter de Gruyter, Berlin, 1985.
[5] A. Laurinčikas, Limit theorems for the Estermann zeta-function. II. Cent. Eur.
J. Math., 3(4) (2005), 580–590.
[6] A. Laurinčikas, R. Macaitienė, Discrete limit theorems for the Estermann zeta-
functions. I. Algebra and Discrete Math. Number 4 (2007), 84-101.
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[7] R. Macaitienė, A joint discrete limit theorem in the space of meromorphic func-
tions for general Dirichlet series. Acta Appl. Math., 97 (2007), 99–112.
[8] H. L. Montgomery, Topics in Multiplicative Number Theory. Springer-Verlag,
Berlin, 1971.
[9] R. Šleževičienė, On some aspects in the theory of the Estermann zeta-function.
Fiz. Mat. Fak. Moksl. Semin. Darb., 5 (2002), 115–130.
Contact information
Antanas
Laurinčikas
Department of Mathematics and Informat-
ics, Vilnius University, Naugarduko 24, LT-
03225 Vilnius, Lithuania
E-Mail: antanas.laurincikas@maf.vu.lt
Renata Macaitienė Department of Mathematics and Informat-
ics, Šiauliai University, P. Visinskio 19, LT-
77156 Siauliai, Lithuania
E-Mail: renata.macaitiene@mi.su.lt
Received by the editors: 26.02.2008
and in final form 14.10.2008.
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