Discrete limit theorems for Estermann zeta-functions. I
A discrete limit theorem in the sense of weak convergence of probability measures on the complex plane for the Estermann zeta-function is obtained. The explicit form of the limit measure in this theorem is given.
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Discrete limit theorems for Estermann zeta-functions. I / A. Laurincikas, R. Macaitiene // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 84–101. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1523842019-06-11T01:25:34Z Discrete limit theorems for Estermann zeta-functions. I Laurincikas, A. Macaitiene, R. A discrete limit theorem in the sense of weak convergence of probability measures on the complex plane for the Estermann zeta-function is obtained. The explicit form of the limit measure in this theorem is given. 2007 Article Discrete limit theorems for Estermann zeta-functions. I / A. Laurincikas, R. Macaitiene // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 84–101. — Бібліогр.: 14 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:11M41. http://dspace.nbuv.gov.ua/handle/123456789/152384 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 on the complex plane for the Estermann zeta-function is obtained. The explicit form of the limit measure in this theorem is given. |
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Article |
| author |
Laurincikas, A. Macaitiene, R. |
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Laurincikas, A. Macaitiene, R. Discrete limit theorems for Estermann zeta-functions. I Algebra and Discrete Mathematics |
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Laurincikas, A. Macaitiene, R. |
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Laurincikas, A. |
| title |
Discrete limit theorems for Estermann zeta-functions. I |
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Discrete limit theorems for Estermann zeta-functions. I |
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Discrete limit theorems for Estermann zeta-functions. I |
| title_fullStr |
Discrete limit theorems for Estermann zeta-functions. I |
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Discrete limit theorems for Estermann zeta-functions. I |
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discrete limit theorems for estermann zeta-functions. i |
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Інститут прикладної математики і механіки НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/152384 |
| citation_txt |
Discrete limit theorems for Estermann zeta-functions. I / A. Laurincikas, R. Macaitiene // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 4. — С. 84–101. — Бібліогр.: 14 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT laurincikasa discretelimittheoremsforestermannzetafunctionsi AT macaitiener discretelimittheoremsforestermannzetafunctionsi |
| first_indexed |
2025-07-13T02:58:11Z |
| last_indexed |
2025-07-13T02:58:11Z |
| _version_ |
1837498890005250048 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2007). pp. 84 – 101
c© Journal “Algebra and Discrete Mathematics”
Discrete limit theorems for Estermann
zeta-functions. I
Antanas Laurinčikas and Renata Macaitienė
Communicated by V. V. Kirichenko
In honour of the 65th birthday of Professor V. V. Kirichenko
Abstract. A discrete limit theorem in the sense of weak
convergence of probability measures on the complex plane for the
Estermann zeta-function is obtained. The explicit form of the limit
measure in this theorem is given.
Introduction
As usual, denote by P, N, N0, Z and C the sets of all primes, posi-
tive integers, non-negative integers, integers, real and complex numbers,
respectively. For arbitrary α ∈ C and m ∈ N, the generalized divisor
function σα(m) is defined by
σα(m) =
∑
d/m
dα.
If α = 0, then σα(m) becomes the divisor function
σ0(m) = d(m) =
∑
d/m
1.
2000 Mathematics Subject Classification: 11M41.
Key words and phrases: compact topological group, Estermann zeta-function,
Haar measure, probability measure, limit theorem, weak convergence.
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.A. Laurinčikas, R. Macaitienė 85
It is well known that, for every positive ǫ,
d(m) ≪ǫ mǫ, m ∈ N.
Here and in the sequel f(x) ≪η g(x) with a positive function g(x), x ∈ I,
means that there exists a constant c = c(η) > 0 such that |f(x)| ≤ cg(x),
x ∈ I. Since
σα(m) = mασ−α(m), (1)
hence we have that
σα(m) ≪ǫ mǫ+max(ℜα,0). (2)
Let s = σ + it be a complex variable, and k and l be coprime integers.
For σ > max(1, 1 + ℜα), the Estermann zeta-function E(s; k
l , α) with
parameters α and k
l is defined by
E
(
s;
k
l
, α
)
=
∞∑
m=1
σα(m)
ms
exp
{
2πim
k
l
}
.
The function E(s; k
l , α) is analytically continuable 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.
The function E(s; k
l , α) with parameter α = 0 was introduced by
T. Estermann in [2] for needs of the representation of a number as the
sum of two products. I. Kiuchi investigated [6] E(s; k
l , α) for α ∈ (−1, 0].
The paper [12] is devoted to zero distribution of the Estermann zeta-
function. The mean-square of E(s; k
l , α) was considered in [14], while the
universality for E(s; k
l , α) was proved in [3]. The mentioned results also
can be found in [13].
In view of [1], we have the functional equation
E
(
s;
k
l
, α
)
= E
(
s − α;
k
l
,−α
)
.
Therefore, without loss of generality, we can suppose that ℜα ≤ 0.
The first attempt to characterize the asymptotic behaviour of the
function E(s; k
l , α) by probabilistic terms was made in [9]. Here a limit
theorem in the sense of weak convergence of probability measures on the
complex plane was proved. To state this theorem, we need some notation.
Let γ = {s ∈ C : |s| = 1} be the unit circle on the complex plane,
and
Ω =
∏
p
γp,
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.86 Discrete limit theorems for Estermann zeta-functions. I
where γp = γ for each prime p. By the Tikhonov theorem, with the
product topology and pointwise multipilication, the infinite-dimensional
torus Ω is a compact topological Abelian group. Therefore, on (Ω,B(Ω)),
where B(S) denotes the class of Borel sets of the space S, the probability
Haar measure mH can be defined, and this leads to a probability space
(Ω,B(Ω), mH). Denote by ω(p) the projection of ω ∈ Ω to the coordinate
space γp, p ∈ P. We extend the function ω(p) to the set N by the formula
ω(m) =
∏
pr‖m
ωr(p), m ∈ N,
where pr ‖ m means that pr | m but pr+1 ∤ m. Now on the probability
space (Ω,B(Ω), mH) we define, for σ > 1
2 , the complex-valued random
element E(σ; k
l , α; ω) by the series
E
(
σ;
k
l
, α; ω
)
=
∞∑
m=1
σα(m)ω(m)
mσ
exp
{
2πim
k
l
}
,
and denote by P C
E,σ its distribution, i.e.,
P C
E,σ(A) = mH
(
ω ∈ Ω : E
(
σ;
k
l
, α; ω
)
∈ A
)
, A ∈ B(C).
Denote by meas{A} the Lebesgue measure of a measurable set A ⊂ R.
Then in [9] the following result has been obtained.
Theorem 1. Suppose that σ > 1
2 and ℜα ≤ 0. Then the probability
measure
1
T
meas
{
t ∈ [0, T ] : E
(
σ + it;
k
l
, α
)
∈ A
}
, A ∈ B(C),
converges weakly to the measure P C
E,σ as T → ∞.
In [10] a generalization of Theorem 1 was given, a limit theorem in
the space of meromorphic functions for the Estermann zeta-function was
obtained. Let D = {s ∈ C : σ > 1
2}, and let M(D) denote the space
of meromorphic on D functions equipped with the topology of uniform
convergence on compacta. Moreover, by H(D) denote the space of an-
alytic on D functions equipped with the topology of M(D). H(D) is a
subspace of M(D). On (Ω,B(Ω), mH), define the H(D)-valued random
element
E
(
s;
k
l
, α; ω
)
=
∞∑
m=1
σα(m)ω(m)
ms
exp
{
2πim
k
l
}
, s ∈ D, ω ∈ Ω,
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.A. Laurinčikas, R. Macaitienė 87
and denote by PH
E its distribution, i.e.,
PH
E (A) = mH
(
ω ∈ Ω : E
(
s;
k
l
, α; ω
)
∈ A
)
, A ∈ B(H(D)).
Then in [10] the following theorem has been proved.
Theorem 2. Suppose that ℜα ≤ 0. Then the probability measure
1
T
meas
{
τ ∈ [0, T ] : E
(
s + iτ ;
k
l
, α
)
∈ A
}
, A ∈ B(M(D)),
converges weakly to PH
E as T → ∞.
Theorems 1 and 2 are of continuous type, the measures in them are
defined by shifts E(σ + it; k
l , α) and E(s + iτ ; k
l , α), when t and τ vary
continuously in the interval [0, T ]. The aim of this paper is to obtain
a discrete limit theorem on the complex plane for the Estermann zeta-
function, when t in E(σ + it; k
l , α) takes values from some discrete set.
Let, for brevity, for N ∈ N0,
µN (...) =
1
N + 1
∑
0≤m≤N
...
1,
where in place of dots a condition satisfied by m is to written.
Theorem 3. Suppose that σ > 1
2 and ℜα ≤ 0. Moreover, let h > 0 be a
fixed number such that exp
{
2πr
h
}
is irrational for all r ∈ Z \ {0}. Then
the probability measure
PN,σ
def
=µN
(
E
(
σ + imh;
k
l
, α
)
∈ A
)
, A ∈ B(C),
converges weakly to P C
E,σ as N → ∞.
1. Limit theorems for absolutely convergent series
Let, for fixed σ1 > 1
2 ,
vn(m) = exp
{
−
(m
n
)σ1
}
.
For n ∈ N and σ > 1
2 , define
En
(
s;
k
l
, α
)
=
∞∑
m=1
σα(m)vn(m)
ms
exp
{
2πim
k
l
}
,
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and, for ω̂ ∈ Ω,
En
(
s;
k
l
, α; ω̂
)
=
∞∑
m=1
σα(m)vn(m)ω̂(m)
ms
exp
{
2πim
k
l
}
.
Since, by (2), for ℜα ≤ 0, the estimate σα(m) ≪ mǫ is true, it is easily
seen that the series for En
(
s; k
l , α
)
and En
(
s; k
l , α; ω
)
converge abso-
lutely in the half-plane σ > 1
2 . The details are similar to those given in
Chapter 5 of [8].
On (C,B(C)), define two probability measures
PN,n,σ = µN
(
En
(
σ + imh;
k
l
, α
)
∈ A
)
and
P̂N,n,σ = µN
(
En
(
σ + imh;
k
l
, α; ω̂
)
∈ A
)
.
Theorem 4. Suppose that σ > 1
2 and ℜα ≤ 0. Let h > 0 be a fixed
number such that exp
{
2πr
h
}
is irrational for all r ∈ Z \ {0}. Then on
(C,B(C)) there exists a probability measure Pn,σ such that the measures
PN,n,σ and P̂N,n,σ both converge weakly to Pn,σ as N → ∞.
The proof of Theorem 4 is based on a discrete limit theorem on the
torus Ω. Define
QN (A) = µN
(
(p−imh : p ∈ P) ∈ A
)
, A ∈ B(Ω).
Lemma 1. Let h > 0 be a fixed number such that exp
{
2πr
h
}
is irrational
for all r ∈ Z \ {0}. Then the probability measure QN converges weakly to
the Haar measure mH as N → ∞.
Proof. The dual group of Ω is
D def
=
⊕
p
Zp,
where Zp = Z for each prime p. An element k = (k2, k3, k5, ...) ∈ D,
where only a finite number of integers kp, p ∈ P, are distinct from zero,
acts on Ω by
ω → ωk =
∏
p
ωkp(p).
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Therefore, the Fourier transform gN (k) of the measure QN is of the form
gN (k) =
∫
Ω
∏
p
ωkp(p)dQN =
1
N + 1
N∑
m=0
∏
p
p−imhkp
=
1
N + 1
N∑
m=0
exp
{
−imh
∑
p
kplog p
}
, (3)
where only a finite number of integers kp, p ∈ P, are distinct from zero.
It is well known that the system {log p : p ∈ P} is linearly independent
over the field of rational numbers Q. Moreover,
∏
p
pkp = exp
{
∑
p
kp log p
}
is a rational number, while, by the hypothesis of the lemma, the number
exp
{
2πr
h
}
is irrational for all r ∈ Z \ {0}. Hence, we obtain that
exp
{
−ih
∑
p
kplogp
}
6= 1
for k 6= 0. Thus, we deduce from (3) that
gN (k) =
1 if k = 0,
1
N+1
1−exp
{
−i(N+1)h
∑
p
kp log p
}
1−exp
{
−ih
∑
p
kp log p
} if k 6= 0.
This shows that
lim
N→∞
gN (k) =
1 if k = 0,
0 if k 6= 0,
and in view of Theorem 1.4.2 of [4] the lemma is proved, since the limit
Fourier transform corresponds the measure mH .
Proof of Theorem 4. Define the function un,σ : Ω → C by the formula
un,σ(ω) =
∞∑
m=1
σα(m)ω(m)vn(m)
mσ
exp
{
2πim
k
l
}
.
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.90 Discrete limit theorems for Estermann zeta-functions. I
Then the function un,σ is continuous, and
un,σ
(
(p−imh : p ∈ P)
)
= En
(
σ + imh;
k
l
, α
)
.
Therefore, PN,n,σ = QNu−1
n,σ. Thus, by Lemma 1 and Theorem 5.1 of
[1] we obtain that the measure PN,n,σ converges weakly to mHu−1
n,σ as
N → ∞.
Now let the function ûn,σ : Ω → C be given by the formula
ûn,σ(ω) =
∞∑
m=1
σα(m)ω̂(m)ω(m)vn(m)
mσ
exp
{
2πim
k
l
}
.
Then, similarly as above, we find that the measure P̂N,n,σ converges
weakly to mH û−1
n,σ as N → ∞. However,
ûn,σ(ω) = un,σ(ωω̂) = un,σ(u(ω)),
where u(ω) = ωω̂, ω ∈ Ω. Hence, mH û−1
n,σ = mH(un,σu)−1 =
(mHu−1)u−1
n,σ = mHu−1
n,σ, since the Haar measure is invariant. �
2. Approximation in the mean
To prove Theorem 3, we have to pass from the function En(s; k
l , α) to
E(s; k
l , α). For this, we need the estimate for the mean
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ + imh;
k
l
, α
)
− En
(
σ + imh;
k
l
, α
)∣∣∣∣ .
If σ > 1
2 and ℜα ≤ 0, then it is known [14] that
T∫
1
∣∣∣∣E
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt ≪ T, T → ∞. (4)
In our case, a discrete version of estimate (4) is necessary. To prove an
estimate of such a kind, we use the Gallagher lemma, see [11], Lemma 1.4.
Lemma 2. Let T0 and T ≥ δ > 0 be real numbers, T be a finite set in
the interval [T0 + δ
2 , T0 + T − δ
2 ], and
Nδ(x) =
∑
t∈T
|t−x|<δ
1.
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Moreover, let S(x) be a complex-valued continuous function on [T0, T0+T ]
having a continuous derivative on (T0, T0 + T ). Then
∑
t∈T
N−1
δ |S(t)|2 ≤ 1
δ
T0+T∫
T0
|S(x)|2dx
+
T0+T∫
T0
|S(x)|2dx
1
2
T0+T∫
T0
|S′(x)|2dx
1
2
.
Lemma 3. Suppose that σ > 1
2 , σ 6= 1, σ 6= 1 + ℜα, if α 6= 0, ℜα ≤ 0
and N → ∞. Then
N∑
m=0
∣∣∣∣E
(
σ + imh + iτ ;
k
l
, α
)∣∣∣∣
2
≪ N + |τ |.
Proof. A simple application of the integral Cauchy formula and (4) show
that
T∫
1
∣∣∣∣E
′
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt ≪ T.
Hence, and from (4), using Lemma 2, we have that
N∑
m=0
∣∣∣∣E
(
σ + imh + iτ ;
k
l
, α
)∣∣∣∣
2
≤ 1
h
hN∫
0
∣∣∣∣E
(
σ + it + iτ ;
k
l
, α
)∣∣∣∣
2
dt
+
hN∫
0
∣∣∣∣E
(
σ + it + iτ ;
k
l
, α
)∣∣∣∣
2
dt
1
2
hN∫
0
∣∣∣∣E
′
(
σ + it + iτ ;
k
l
, α
)∣∣∣∣
2
dt
1
2
≪
hN+|τ |∫
−|τ |
∣∣∣∣E
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt
+
hN+|τ |∫
−|τ |
∣∣∣∣E
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt
1
2
hN+|τ |∫
−|τ |
∣∣∣∣E
′
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt
1
2
≪ N + |τ |.
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Theorem 5. Suppose that σ > 1
2 and ℜα ≤ 0. Then
lim
n→∞
lim sup
N→∞
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ + imh;
k
l
, α
)
− En
(
σ + imh;
k
l
, α
)∣∣∣∣ = 0.
Proof. Let σ1 the same as in Section 1. For n ∈ N, define
ln(s) =
s
σ1
Γ
(
s
σ1
)
ns.
Then, see [9], for σ > 1
2 ,
En
(
s;
k
l
, α
)
=
1
2πi
σ1+i∞∫
σ1−i∞
E
(
s + z;
k
l
, α
)
ln(z)
dz
z
.
Define σ2 by
σ > σ2 >
1
2 if α = 0 or 1 + ℜα − σ > 0,
1 + ℜα otherwise.
Thus, we obtain by the residue theorem that
En
(
s;
k
l
, α
)
=
1
2πi
σ2−σ+i∞∫
σ2−σ−i∞
E
(
s + z;
k
l
, α
)
ln(z)
dz
z
+E
(
s;
k
l
, α
)
+ R
(
s;
k
l
, α
)
,
where
R
(
s;
k
l
, α
)
=
Res
z=1−s
E(s + z; k
l , α) ln(z)
z if α = 0,
Res
z=1−s
E(s + z; k
l , α) ln(z)
z + Res
z=1+α−s
E(s + z; k
l , α) ln(z)
z
if 1 + ℜα − σ > 0.
Hence, we have
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ + imh;
k
l
, α
)
− En
(
σ + imh;
k
l
, α
)∣∣∣∣
≪
∞∫
−∞
(
|ln(σ2 − σ + iτ)|
|σ2 − σ + iτ |
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ2 + imh + iτ ;
k
l
, α
)∣∣∣∣
)
dτ
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+
1
N + 1
N∑
m=0
∣∣∣∣R
(
σ2 − σ + imh;
k
l
, α
)∣∣∣∣ . (5)
We can choose σ2 6= 1 and σ2 6= 1 + ℜα. Thus, by Lemma 3
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ2 + imh + iτ ;
k
l
, α
)∣∣∣∣
≪ 1
N
(
N∑
m=0
1
) 1
2
(
N∑
m=0
∣∣∣∣E
(
σ2 + imh + iτ ;
k
l
, α
)∣∣∣∣
2
) 1
2
≪ 1 + |τ |. (6)
Applying Lemma 2 again, we find that
N∑
m=0
∣∣∣∣R
(
σ2 − σ + imh;
k
l
, α
)∣∣∣∣
≪
√
N
(
N∑
m=0
∣∣∣∣R
(
σ2 − σ + imh;
k
l
, α
)∣∣∣∣
2
) 1
2
≪
√
N
( Nh∫
0
∣∣∣∣R
(
σ2 − σ + it;
k
l
, α
)∣∣∣∣
2
dt
+
( Nh∫
0
∣∣∣∣R
(
σ2 − σ + it;
k
l
, α
)∣∣∣∣
2
dt
) 1
2
( Nh∫
0
∣∣∣∣R
′
(
σ2 − σ + it;
k
l
, α
)∣∣∣∣
2
dt
) 1
2
) 1
2
.
(7)
Since the function ln(s) contains the Euler gamma-function, we obtain
the estimate
Nh∫
0
∣∣∣∣R
(
σ2 − σ + it;
k
l
, α
)∣∣∣∣
2
dt ≪ 1. (8)
This and application of the Cauchy integral formula give the bound
Nh∫
0
∣∣∣∣R
′
(
σ2 − σ + it;
k
l
, α
)∣∣∣∣
2
dt ≪ 1.
This and (7), (8) lead to the estimate
1
N + 1
N∑
m=0
∣∣∣∣R
(
σ2 − σ + imh;
k
l
, α
)∣∣∣∣
2
dt ≪ 1√
N
.
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.94 Discrete limit theorems for Estermann zeta-functions. I
Therefore, in view of (5) and (6)
lim
n→∞
lim sup
N→∞
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ + imh;
k
l
, α
)
− En
(
σ + imh;
k
l
, α
)∣∣∣∣
≪ lim
n→∞
∞∫
−∞
|ln(σ2 − σ + iτ)| (1 + |τ |)dt. (9)
However, since σ2 − σ < 0,
lim
n→∞
∞∫
−∞
|ln(σ2 − σ + iτ)| (1 + |τ |)dt = 0,
and the theorem is a consequence of estimate (9).
We also need an analogue of Theorem 5 for the functions E(s; k
l , α; ω)
and En(s; k
l , α; ω)
Theorem 6. Let σ > 1
2 and ℜα ≤ 0. Then, for almost all ω ∈ Ω,
lim
n→∞
lim sup
N→∞
1
N + 1
N∑
m=0
∣∣∣∣E
(
σ + imh;
k
l
, α; ω
)
−En
(
σ + imh;
k
l
, α; ω
)∣∣∣∣ = 0.
Proof. In [9], Lemma 5, it was obtained that, under the hypotheses of
the theorem,
T∫
0
∣∣∣∣E
(
σ + it;
k
l
, α; ω
)∣∣∣∣
2
dt ≪ T
for almost all ω ∈ Ω. Hence, similarly to the proof of Lemma 3, we obtain
that
N∑
m=0
∣∣∣∣E
(
σ + imh + iτ ;
k
l
, α; ω
)∣∣∣∣
2
≪ N + |τ | (10)
for almost all ω ∈ Ω.
The random variables ω(m), m ∈ N, are pointwise orthogonal, that
is
E
(
ω(m)ω(n)
)
=
1 if m = n,
0 if m 6= n,
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where E(X) denotes the expectation of X. Hence, we have that
E
(
σα(m)ω(m)
mσ
σα(n)ω(n)
nσ
exp
{
2πi
k
l
(m − n)
})
=
|σα(m)|2
m2σ if m = n,
0 if m 6= n.
Thus, in view of (2), the series
∞∑
m=1
E
∣∣∣∣
σα(m)ω(m)
mσ
exp
{
2πim
k
l
}∣∣∣∣
2
log2 m
converges for any fixed σ > 1
2 . Therefore, by the Rademacher theorem,
see, for example [11], the series, for any fixed σ > 1
2 ,
∞∑
m=1
σα(m)ω(m)
mσ
exp
{
2πim
k
l
}
converges for almost all ω ∈ Ω. Hence, the series
∞∑
m=1
σα(m)ω(m)
mσ
exp
{
2πim
k
l
}
,
for almost all ω ∈ Ω, converges uniformly on compact subsets of the
half-plane {s ∈ C : σ > 1
2}. This shows that, for almost all ω ∈ Ω,
the function E(s; k
l , α; ω) is analytic in the region {s ∈ C : σ > 1
2}.
Therefore, using the representation
En
(
s;
k
l
, α; ω
)
=
1
2πi
σ1+i∞∫
σ1−i∞
E
(
s + z;
k
l
, α; ω
)
ln(z)
dz
z
,
we obtain that, for 1
2 < σ2 < σ,
En
(
s;
k
l
, α; ω
)
=
1
2πi
σ2−σ+i∞∫
σ2−σ−i∞
E
(
s+z;
k
l
, α; ω
)
ln(z)
dz
z
+E
(
s;
k
l
, α; ω
)
for almost all ω ∈ Ω. Using the latter formula and (9), we complete the
proof in the same way as in the case of Theorem 5.
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.96 Discrete limit theorems for Estermann zeta-functions. I
3. Proof of Theorem 3
Define one more probability measure
P̂N,σ = µN
(
E
(
σ + imh;
k
l
, α; ω
)
∈ A
)
, A ∈ B(C).
We begin the proof of Theorem 3 with the following statement.
Theorem 7. Suppose that σ > 1
2 and ℜα ≤ 0. Then on (C,B(C)) there
exists a probability measure Pσ such that the measures PN,σ and P̂N,σ both
converge weakly to Pσ as N → ∞.
Proof. By Theorem 4, for σ > 1
2 , the measures PN,n,σ
P̂N,n,σ = µN
(
En
(
σ + imh;
k
l
, α; ω
)
∈ A
)
, A ∈ B(C),
for every ω ∈ Ω, both converge weakly to the same measure Pn,σ as
N → ∞.
For any positive M , by the Chebyshev inequality
PN,n,σ
(
{z ∈ C : |z| > M}
)
= µN
(∣∣∣∣En
(
σ + imh;
k
l
, α
)∣∣∣∣ > M
)
≤ 1
M(N + 1)
N∑
m=0
∣∣∣∣En
(
σ + imh;
k
l
, α
)∣∣∣∣ .
(11)
As we have observed above, the series for En(s; k
l , α) converges absolutely
for σ > 1
2 . Also, the latter property holds for E′
n(s; k
l , α). Therefore, for
σ > 1
2 ,
lim
T→∞
1
T
T∫
1
∣∣∣∣En
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt =
∞∑
m=1
|σα(m)|2v2
n(m)
m2σ
≤
∞∑
m=1
|σα(m)|2
m2σ
< ∞, (12)
and
lim
T→∞
1
T
T∫
1
∣∣∣∣E
′
n
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt =
∞∑
m=1
|σα(m)|2v2
n(m)log2m
m2σ
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≤
∞∑
m=1
|σα(m)|2log2m
m2σ
< ∞.(13)
An application of Lemma 2 yields
1
N + 1
N∑
m=0
∣∣∣∣En
(
σ + imh;
k
l
, α
)∣∣∣∣≪
1√
N
(
N∑
m=0
∣∣∣∣En
(
σ + imh;
k
l
, α
)∣∣∣∣
2
) 1
2
≪ 1√
N
(
1
Nh
Nh∫
0
∣∣∣∣En
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt
+
(
1
N
Nh∫
0
∣∣∣∣En
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt
) 1
2
(
1
N
hN∫
0
∣∣∣∣E
′
n
(
σ + it;
k
l
, α
)∣∣∣∣
2
dt
) 1
2
) 1
2
.
This, (12) and (13) show that
sup
n∈N
lim sup
N→∞
1
N + 1
N∑
m=0
∣∣∣∣En
(
σ + imh;
k
l
, α; ω
)∣∣∣∣ ≤ C(h)R, (14)
where
R =
∞∑
m=1
|σα(m)|2
m2σ
+
(
∞∑
m=1
|σα(m)|2
m2σ
) 1
2
(
∞∑
m=1
|σα(m)|2log2m
m2σ
) 1
2
1
2
< ∞.
For arbitrary ǫ > 0, let Mǫ = C(h)Rǫ−1. Then, taking into account (11)
and (14), we find that
lim sup
N→∞
PN,n,σ
(
{z ∈ C : |z| > Mǫ}
)
≤ ǫ. (15)
The function u : C → R, z → |z|, is continuous. Therefore, by Theorem 4
and Theorem 5.1 of [1] we have that, for σ > 1
2 , the probability measure
µN
(∣∣∣∣En
(
σ + imh;
k
l
, α
)∣∣∣∣ ∈ A
)
, A ∈ B(R),
converges weakly to Pn,σu−1 as N → ∞. This together with Theorem 2.1
of [1] and (15) implies
Pn,σ
(
{z ∈ C : |z| > Mǫ}
)
≤ lim inf
N→∞
PN,n,σ
(
{z ∈ C : |z| > Mǫ}
)
≤ lim sup
N→∞
PN,n,σ
(
{z ∈ C : |z| > Mǫ}
)
≤ ǫ
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(16)
for all n ∈ N. Define Kǫ = {z ∈ C : |z| ≤ Mǫ}. Then the set Kǫ is
compact, and by (16)
Pn,σ(Kǫ) ≥ 1 − ǫ
for all n ∈ N. This means that the family of probability measures {Pn,σ :
n ∈ N} is tight, and by the Prokhorov theorem, see Theorem 6.1 of [1],
it is relatively compact. Therefore, there exists a subsequence {Pnk,σ} ⊂
{Pn,σ} such that Pnk,σ converges weakly to some measure Pσ on (C,B(C))
as k → ∞.
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(σ) = En
(
σ + iθN ;
k
l
, α
)
and denote by Xn = Xn(σ) the complex-valued random variable with the
distribution Pn,σ. Then by Theorem 4
XN,n
D−→
N→∞
Xn, (17)
where
D−→ denotes the convergence in distribution. Moreover, from the
above remark
Xnk
(σ)
D−→
k→∞
Pσ. (18)
Define
XN (σ) = E
(
σ + iθN ;
k
l
, α
)
.
Then in view of Theorem 5, for σ > 1
2 and any ǫ > 0,
lim
n→∞
lim sup
N→∞
P (|XN (σ) − XN,n(σ)| ≥ ǫ)
= lim
n→∞
lim sup
N→∞
µN
(∣∣∣∣E
(
σ + imh;
k
l
, α
)
− En
(
σ + imh;
k
l
, α
)∣∣∣∣ ≥ ǫ
)
≤ lim
n→∞
lim sup
N→∞
1
ǫ(N + 1)
∞∑
m=1
∣∣∣∣E
(
σ + imh;
k
l
, α
)
− En
(
σ + imh;
k
l
, α
)∣∣∣∣ = 0.
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This, (17), (18) and Theorem 4.2 of [1] show that
XN (σ)
D−→
N→∞
Pσ, (19)
and this is equivalent to weak convergence of PN,σ to Pσ as N → ∞.
Relation (19) shows that the measure Pσ is independent of the choice
of the sequence Pnk,σ. Hence, we obtain that
Xn(σ)
D−→
n→∞
Pσ. (20)
Now define
X̂N,n = X̂N,n(σ) = En
(
σ + iθN ;
k
l
, α; ω
)
and
X̂N = X̂N (σ) = E
(
σ + iθN ;
k
l
, α; ω
)
.
Then in the same way as above, using (20) and Theorem 6, we find that
the measure P̂N,σ also converges weakly to Pσ as N → ∞.
Proof of Theorem 3. In view of Theorem 7, it remains to identify the
limit measure Pσ.
Let A ∈ B(C) be a fixed continuity set of the limit measure Pσ in
Theorem 7. Then we have that
lim
N→∞
µN
(
E
(
σ + imh;
k
l
, α
)
∈ A
)
= Pσ(A). (21)
Now on (Ω,B(Ω)) define the random variable θ by the formula
θ = θ(ω) =
{
1 if E
(
σ; k
l , α; ω
)
∈ A,
0 if E
(
σ; k
l , α; ω
)
/∈ A.
Then we have that
Eθ =
∫
Ω
θdmH = mH
(
ω ∈ Ω : E
(
s;
k
l
, α; ω
)
∈ A
)
= P C
E,σ. (22)
Let ah = {p−ih : p ∈ P}. Define the transformation fh on Ω by
fh(ω) = ahω, ω ∈ Ω. Then fh is a measurable measure preserving trans-
formation on (Ω,B(Ω), mH). In [5] it was obtained that the transforma-
tion fh is ergodic. Then by the classical Birkhoff-Khinchine theorem, see,
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for example [7], we obtain that
lim
N→∞
1
N + 1
N∑
m=0
θ
(
fm
h (ω)
)
= Eθ (23)
for almost all ω ∈ Ω. However, by the definition of fh, we have that
1
N + 1
N∑
m=0
θ
(
fm
h (ω)
)
= µN
(
E
(
σ + imh;
k
l
, α; ω
)
∈ A
)
.
From this, (22) and (23) we obtain that
lim
N→∞
µN
(
E
(
σ + imh;
k
l
, α; ω
)
∈ A
)
= P C
E,σ(A).
Therefore, by (21), Pσ(A) = P C
E,σ(A). Since A is arbitrary continuity
set of Pσ, the latter equality is true for any continuity set A. However,
all continuity sets constitute the determining class, and we have that
Pσ(A) = P C
E,σ(A) for all A ∈ B(C). The theorem is proved. �
References
[1] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
[2] T. Estermann, On the representation of a number as the sum of two products,
Proc. London Math. Soc., N.31, 1930, pp.123–133.
[3] R. Garunkštis, A. Laurinčikas, R. Šleževičienė, J. Steuding, On the universality of
Estermann zeta-functions, Analysis, N.22, 2002, pp.285–296.
[4] H. Heyer, Probability Measures on Locally Compact Groups. Springer-Verlag,
Berlin, 1977.
[5] R. Kačinskaitė, A discrete limit theorem for the Matsumoto zeta-function on the
complex plane, Lith. Math. J., N.40(4), 2000, pp. 364–378.
[6] I. Kiuchi, On an exponentials sum involving the arithmetic function σa(n),
Math. J. Okayama Univ., N.29, 1987, pp.193–205.
[7] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985.
[8] A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer Academic
Publishers, Dordrecht, Boston, London, 1996.
[9] A. Laurinčikas, Limit theorems for the Estermann zeta-function. I, Statist. Probab.
Lett., N.72, 2005, pp.227–237.
[10] A. Laurinčikas, Limit theorems for the Estermann zeta-function. II, Cent. Eur.
J. Math., N.3(4), 2005, pp.580–590.
Jo
u
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al
A
lg
eb
ra
D
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cr
et
e
M
at
h
.A. Laurinčikas, R. Macaitienė 101
[11] H. L. Montgomery, Topics in Multiplicative Number Theory, Springer-Verlag,
Berlin, 1971.
[12] R. Šleževičienė, J. Steuding, On the zeros of the Estermann zeta-function, Integral
Transforms and Special Functions, N.13, 2002, pp.363–371.
[13] R. Šleževičienė, On some aspects in the theory of the Estermann zeta-function,
Fiz. Mat. Fak. Moksl. Semin. Darb., N.5, 2002, pp.115–130.
[14] R. Šleževičienė, J. Steuding, The mean–square of the Estermann zeta-functon,
Faculty of Mathematics and Informatics, Vilnius University, Preprint 2002-32,
2002.
Contact information
A. Laurinčikas Department of Mathematics
and Informatics,
Vilnius University,
Naugarduko 24,
LT-03225 Vilnius,
Lithuania
E-Mail: antanas.laurincikas@mif.vu.lt
R. Macaitienė Department of Mathematics
and Informatics,
Šiauliai University,
P. Vǐsinskio 19,
LT-77156 Šiauliai,
Lithuania
E-Mail: renata.macaitiene@mi.su.lt
Received by the editors: 09.07.2007
and in final form 07.04.2008.
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