Value distribution of general Dirichlet series. VIII
A joint limit theorem on the complex plane for a new class of general Dirichlet series is proved.
Збережено в:
| Дата: | 2006 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2006
|
| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/157393 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Value distribution of general Dirichlet series. VIII / A. Laurincikas // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 40–56. — Бібліогр.: 20 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-157393 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1573932019-06-21T01:28:28Z Value distribution of general Dirichlet series. VIII Laurincikas, A. A joint limit theorem on the complex plane for a new class of general Dirichlet series is proved. 2006 Article Value distribution of general Dirichlet series. VIII / A. Laurincikas // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 40–56. — Бібліогр.: 20 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 11M41. http://dspace.nbuv.gov.ua/handle/123456789/157393 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A joint limit theorem on the complex plane for a
new class of general Dirichlet series is proved. |
| format |
Article |
| author |
Laurincikas, A. |
| spellingShingle |
Laurincikas, A. Value distribution of general Dirichlet series. VIII Algebra and Discrete Mathematics |
| author_facet |
Laurincikas, A. |
| author_sort |
Laurincikas, A. |
| title |
Value distribution of general Dirichlet series. VIII |
| title_short |
Value distribution of general Dirichlet series. VIII |
| title_full |
Value distribution of general Dirichlet series. VIII |
| title_fullStr |
Value distribution of general Dirichlet series. VIII |
| title_full_unstemmed |
Value distribution of general Dirichlet series. VIII |
| title_sort |
value distribution of general dirichlet series. viii |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157393 |
| citation_txt |
Value distribution of general Dirichlet series. VIII / A. Laurincikas // Algebra and Discrete Mathematics. — 2006. — Vol. 5, № 4. — С. 40–56. — Бібліогр.: 20 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT laurincikasa valuedistributionofgeneraldirichletseriesviii |
| first_indexed |
2025-07-14T09:49:48Z |
| last_indexed |
2025-07-14T09:49:48Z |
| _version_ |
1837615377480155136 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2006). pp. 40 – 56
c© Journal “Algebra and Discrete Mathematics”
Value distribution of general Dirichlet series.
VIII
A. Laurinčikas
Communicated by V. V. Kirichenko
Abstract. A joint limit theorem on the complex plane for a
new class of general Dirichlet series is proved.
1. Introduction
Let s = σ + it be a complex variable, {am : m ∈ N} be a sequence of
complex numbers, and let {λm : m ∈ N} be an increasing sequence of
positive numbers, lim
m→∞
λm = +∞. The series of the form
f(s) =
∞∑
m=1
ame−λms (1)
is called a general Dirichlet series. If λm = log m, we obtain the ordinary
Dirichlet series
∞∑
m=1
am
ms
.
It is well known that the region of convergence as well as of absolute
convergence of Dirichlet series is a half-plane.
The first probabilistic results for Dirichlet series were obtained by
H.Bohr and B.Jessen. In [2] and [3] they proved theorems for the Rie-
mann zeta-function which are similar to modern limit theorems in the
sense of weak convergence of probability measures. The investigations
2000 Mathematics Subject Classification: 11M41.
Key words and phrases: Compact topological group, general Dirichlet series,
Haar measure, limit theorem, probability measure, random element, weak convergence.
A. Laurinčikas 41
of H.Bohr and B.Jessen were developed and generalized by A. Wint-
ner, V.Borchsenius, A.Selberg, P.D.T.A. Elliott, A.Ghosh, K.Matsumoto,
B.Bagchi, E.M.Nikishin, E.Stankus, J.Steuding, W.Schwarz, the author
and others. The results of such a kind can be found in [7], [8], [14] and
[20].
Limit theorems in the sense of weak convergence of probability mea-
sures in various spaces for general Dirichlet series were obtained [4]-[6],
[10]-[14] and [18], [19]. Limit theorems on the complex plane for gen-
eral Dirichlet series were proved in [12]-[14]. Denote by meas{A} the
Lebesgue measure of a measurable set A ∈ R, and let, for T > 0,
νT (...) =
1
T
meas{t ∈ [0, T ] : ...},
where in place of dots a condition satisfied by t is to be written. Moreover,
let B(S) be the class of Borel sets of the space S.
Denote by γ the unit circle {s ∈ C : |s| = 1} on the complex plane
C, and define
Ω =
∞∏
m=1
γm,
where γm = γ for each m ∈ N. Then the infinite-dimensional torus
Ω in view of the Tikhonov theorem is a compact topological Abelian
group, therefore the probability Haar measure mH on (Ω,B(Ω)) can be
defined. This gives a probability space (Ω,B(Ω), mH). Denote by ω(m)
the projection of ω ∈ Ω to the coordinate space γm, m ∈ N.
Suppose that the series (1) converges absolutely for σ > σa. Then the
function f(s) is analytic in the half-plane {s ∈ C : σ > σa}. Moreover,
we require that the function f(s) should be meromorphically continuable
to the half-plane {s ∈ C : σ > σ1}, σ1 < σa, all poles being included in a
compact set, and that, for σ > σ1, the estimates
f(σ + it) ≪ |t|α, α > 0, |t| > t0 > 0, (2)
and
T∫
−T
|f(σ + it)|2 d t ≪ T, T → ∞, (3)
should be satisfied. Suppose that the exponents λm satisfy the inequality
λm > (log m)δ (4)
with some positive δ > 0. Then in [12] it was proved that under the last
conditions, for σ > σ1,
f(σ, ω) =
∞∑
m=1
amω(m)e−λmσ,
42 Value distribution of general Dirichlet series. VIII
is a complex-valued random variable defined on the probability space
(Ω,B(Ω), mH). If the system {λm} is linearly independent over the field
of rational numbers, then it was obtained in [12] that, for σ > σ1, the
probability measure
νT
(
f(σ + it) ∈ A
)
, A ∈ B(C), (6)
converges weakly to the distribution of the random variable f(σ, ω) as
T → ∞.
Condition (4) is rather strong, it limits a class of Dirichlet series for
which a limit theorem is true. Suppose that, for σ > σ1,
∞∑
m=1
|am|2e−2λmσlog2m < ∞. (7)
Then in [14] the following statement has been obtained.
Theorem A. Suppose that the system {λm} is linearly independent over
the field of rational numbers, and conditions (2), (3) and (7) are satisfied.
Then the probability measure (6) converges weakly to the distribution of
the random element f(σ, ω) as T → ∞.
In [13] a joint limit theorem on the complex plane for general Dirichlet
series was proved. Let, for σ > σaj ,
fj(s) =
∞∑
m=1
amje
−λmjs,
where {amj} and {λmj} are a sequence of complex numbers and an in-
creasing sequence of positive numbers, lim
m→∞
λmj = +∞, respectively,
j = 1, ..., r, r > 1. Suppose that the function fj(s) is meromorphically
continuable to the region {s ∈ C : σ > σ1j}, σ1j < σaj , j = 1, ..., n, all
poles being included in a compact set, and, for σ > σ1j , the estimates
fj(σ + it) ≪ |t|αj , αj > 0, |t| > t0 > 0, (8)
and
T∫
−T
|fj(σ + it)|2 d t ≪ T, T → ∞, (9)
j = 1, ..., r, are satisfied. Moreover, we assume that λmj = λm, j =
1, ..., r, and
λm > c(log m)δ, c > 0, δ > 0. (10)
A. Laurinčikas 43
Let C
r = C × ... × C︸ ︷︷ ︸
r
. On the probability space (Ω,B(Ω), mH) define, for
σ1 > σ11, ..., σr > σ1r, an C
r-valued random element F = F (σ1, ..., σr; ω)
by
F = F (σ1, ..., σr, ω) = (f1(σ1, ω), ..., fn(σr, ω)) ,
where
fj(σj , ω) =
∞∑
m=1
amjω(m)e−λmσj , ω ∈ Ω.
Theorem B [13]. Suppose that the system {λm} is linearly indepen-
dent over the field of rational numbers, and that conditions (8)-(10) are
satisfied. Then the probability
PT (A)
def
= νT
(
(f1(σ1 + it), ..., fr(σr + it)) ∈ A
)
, A ∈ B(Cr),
for σ1 > σ1j , ..., σr > σ1r, converges weakly to the distribution of the
random element F (σ1, ..., σr; ω) as T → ∞.
The aim of this note is to change condition (10) in Theorem B by
a weaker one and to consider a general case of different exponents λmj .
Therefore, for the proof we will apply a method different from that of
[13]. Suppose that, for σj > σ1j ,
∞∑
m=1
|amj |
2e−2λmjσj log2m < ∞, j = 1, ..., r. (11)
Moreover, define Ωr = Ω1 × ... × Ωr where Ωj = Ω for j = 1, ..., r. Then
Ωr is also a compact topological Abelian group. Denote by mHr the
probability Haar measure on (Ωr,B(Ωr)).
In the next section it will be proved that, under condition (11), for
σ1 > σ11, ..., σr > σ1r,
F (σ1, ..., σr; ω) = (f1(σ1, ω1), ..., fr(σr, ωr)),
where
fj(σj , ωj) =
∞∑
m=1
amjωj(m)e−λmjσj , ωj ∈ Ωj , j = 1, ..., r, ω = (ω1, ..., ωr),
is a C
n-valued random element defined on the probability
space (Ωr,B(Ωr), mHr).
Theorem 1. Suppose that the set
r⋃
j=1
∞⋃
m=1
{λmj} is linearly independent
over the field of rational numbers, and that conditions (8), (9) and (11)
44 Value distribution of general Dirichlet series. VIII
are satisfied. Then the probability measure PT converges weakly to the
distribution of the random element F (σ1, ..., σr; ω) as T → ∞.
Note that joint limit theorems can be used to derive the joint univer-
sality for considered functions, see, for example, [16] and [17].
2. The random element F (σ1, ..., σr; ω)
In this section we will prove that, under condition (11), F (σ1, ..., σr; ω) is
a C
r-valued random element. For the proof, we will apply a Rademacher’s
theorem on series of pairwise orthogonal random variables. Denote by Eξ
the expectation of the random element ξ.
Lemma 2.[20] Suppose that {Xn} is a sequence of orthogonal random
variables such that
∞∑
m=1
E|Xm|2log2m < ∞.
Then the series
∞∑
m=1
Xm
converges almost surely.
Theorem 3. Suppose that condition (11) holds. Then F (σ1, ..., σr; ω),
for σ1 > σ11, ..., σr > σ1r, is a C
r-valued random element defined on the
probability space (Ω,B(Ω), mHr).
Proof. Clearly, it suffices to prove that, for each j = 1, ..., r,
fj(σj , ω) =
∞∑
m=1
amjω(m)e−λmjσj , ω ∈ Ω,
for σj > σ1j , is a complex-valued random variable on the probability
space (Ω,B(Ω), mH).
We fix j ∈ {1, ..., r}. Let σ > σ1j be fixed, and
ξmj = ξmj(ω) = amjω(m)e−λmjσ.
Then {ξmj} is a sequence of pairwise orthogonal complex-valued random
variables defined on the probability space (Ω,B(Ω), mH). Really, denot-
ing by z the complex conjugate of z ∈ C, we find
E(ξmj , ξkj) =
∫
Ω
ξmj(ω)ξkj(ω)dmH = amjakje
−(λmj+λkj)σ
∫
Ω
ω(m)ω(k)dmH
=
{
0 if m 6= k,
|amj |
2e−2λmjσ if m = k.
A. Laurinčikas 45
Since σ > σ1j , hence we have in view of (11) that
∞∑
m=1
E|ξmj |
2log2m < ∞.
This and Lemma 2 show that the series
∞∑
m=1
ξmj =
∞∑
m=1
amjω(m)e−λmjσ = f(σ, ω) (12)
converges almost surely with respect the Haar measure mH . Then
( ∞∑
m=1
am1ω1(m)e−λm1σ1 , ...,
∞∑
m=1
amrωr(m)e−λmrσr
)
converges almost surely in C
r, and this proves the theorem. We note that
mHr = mH × ... × mH︸ ︷︷ ︸
r
.
3. Joint limit theorems for Dirichlet polynomials
We start with a joint limit theorem on the torus Ωr. Define the probability
measure
QT,r(A) = νT
((
(eitλm1 : m ∈ N), ..., (eitλmr : m ∈ N)
)
∈ A
)
.
Lemma 4. The probability measure QT,r converges weakly to the
Haar measure mHr on (Ωr,B(Ωr)) as T → ∞.
Proof. The dual group of Ωr is
r⊕
j=1
∞⊕
m=1
Zmj ,
where Zmj = Z for all m ∈ N and j = 1, ..., r.
(k1, ..., kr) = (k11, k21, ..., k1r, k2r, ...) ∈
r⊕
j=1
∞⊕
m=1
Zmj ,
where only a finite number of integers kmj , m ∈ N, j = 1, ..., r, are
distinct from zero, acts on Ωr by
(x1, ..., xr) → (x
k
1
1 , ..., x
kr
r ) =
r∏
j=1
∞∏
m=1
x
kmj
mj , xj = (x1j , x2j , ...), xmj ∈ γ,
46 Value distribution of general Dirichlet series. VIII
m ∈ N, j = 1..., r. Therefore, the Fourier transform gT,r(k1, ..., kr) of the
measure QT,r is
gT,r(k1, ..., kr) =
∫
Ωr
r∏
j=1
∞∏
m=1
x
kmj
mj dQT,r =
1
T
∫ T
0
r∏
j=1
∞∏
m=1
eitkmjλmj d t
=
1
T
∫ T
0
exp{it
r∑
j=1
∞∑
m=1
kmjλmj}d t.
Since the set
⋃r
j=1
⋃
∞
m=1{λmj} is linearly independent over the field of
rational numbers, hence we find that
gT,r(k1, ..., kr) =
1 if (k1, ..., kr) = (0, ..., 0),
exp
{
iT
r∑
j=1
∞∑
m=1
kmjλmj
}
−1
iT
r∑
j=1
∞∑
m=1
kmjλmj
if (k1, ..., kr) 6= (0, ..., 0).
Therefore,
lim
T→∞
gT,r(k1, ..., kr) =
{
1 if (k1, ..., kr) = (0, ..., 0),
0 if (k1, ..., kr) 6= (0, ..., 0).
This and continuity theorems for probability measures on compact groups
[7] show that the probability measure QT,r converges weakly to the Haar
measure mHr as T → ∞.
Let σ2j > σaj − σ1j , and, for m, n ∈ N,
vj(m, n) = exp{−e(λm−λn)σ2j}, j = 1, ..., r.
Define, for Nj ∈ N, σj > σ1j and ω̂j ∈ Ω,
fNj ,j,n(σj + it) =
Nj∑
m=1
amjvj(m, n)e−λmj(σj+it),
fNj ,j,n(σj + it, ω̂j) =
Nj∑
m=1
amjω̂j(m)vj(m, n)e−λmj(σj+it), j = 1, ..., r,
and consider the weak convergence of the probability measures
PT,N1,...,Nr,n(A) = νT
(
(fN1,1,n(σ1 + it), ..., fNr,r,n(σr + it)
)
∈ A
and
P̂T,N1,...,Nr ,n(A) = νT
(
(fN1,1,n(σ1 + it, ω̂1), ..., fNr,r,n(σr + it, ω̂r)
)
∈ A,
A. Laurinčikas 47
where (ω̂1, ..., ω̂r) ∈ Ωr and A ∈ B(Cr).
Theorem 5. The probability measures PT,N1,...,Nr ,n and P̂T,N1,...,Nr,n
both converge weakly to the same probability measure on (Cr,B(Cr)) as
T → ∞.
Proof. Let a function h : Ωr → C
r be given by
h(ω1, ..., ωr) =
( N1∑
m=1
am1v(m, n)e−λm1σ1ω−1
1 (m), ...,
Nr∑
m=1
amrv(m, n)e−λmrσrω−1
r (m)
)
,
(ω1, ..., ωr) ∈ Ωr. Then, clearly,
h
(
(eitλm1 : m ∈ N), ..., (eitλmr : m ∈ N)
)
=
(
fN1,1,n(σ1 + it), ..., fNr,r,n(σr + it)
)
def
= fN1,...,Nr ,n(σ1, ..., σr; t),
and the function h is continuous. Therefore, PT,N1,...,Nr,n = QT,rh
−1, and
by Theorem 5.1 of [1] and Lemma 4 the probability measure PT,N1,...,Nr,n
converges weakly to mHrh
−1 as T → ∞.
Now let h1 : Ωr → Ωr be defined by the formula
h1(ω1, ..., ωr) = (ω1ω̂
−1
1 , ..., ωrω̂
−1
r ).
Then we have that
(
fN1,1,n(σ1 + it, ω̂1), ..., fNr,1,n(σr + it, ω̂r)
)
=
h
(
h1
(
(eitλm1 : m ∈ N), ..., (eitλmr : m ∈ N)
))
.
Similarly to the case of the measure PT,N1,...,Nr,n we obtain that the proba-
bility measure PT,N1,...,Nr,n converges weakly to the measure mHr(hh1)
−1
as T → ∞. The Haar measure mHr is invariant with respect to transla-
tions by points from Ωr. Therefore,
mHr(hh1)
−1 = (mHrh
−1
1 )h−1 = mHrh
−1,
and the theorem is proved.
48 Value distribution of general Dirichlet series. VIII
4. Limit theorems for absolutely convergent series
Define, for ωj ∈ Ω and j = 1, ..., r,
fn,j(s) =
∞∑
m=1
amjvj(m, n)e−λmjs
and
fn,j(s, ωj) =
∞∑
m=1
amjωj(m)vj(m, n)e−λmjs.
Then the latter series both converge absolutely for σ > σ1j . The proof
of this is given in [12], Lemma 4. In this section we consider the weak
convergence of the probability measures
PT,n(A) = νT
(
((fn,1(σ1 + it), ..., fn,r(σr + it)) ∈ A
)
, A ∈ B(Cr),
and
P̂T,n(A) = νT
(
((fn,1(σ1 + it, ω1), ..., fn,r(σr + it, ωr)) ∈ A
)
, A ∈ B(Cr).
Theorem 6. Let σj > σ1j, j = 1, ..., r. Then there exists a probability
measure Pn on (Cr,B(C)) such that the measures PT,n and P̂T,n both
converge weakly to Pn as T → ∞.
Proof. We will apply Theorem 5. Without loss of generality
we take N1 = ... = Nr
def
= N . Then by Theorem 5 the mea-
sures PT,N1,...,Nr ,n
def
= PT,N,n and P̂T,N1,...,Nr,n
def
= P̂T,N,n both converge
weakly to the same measure PN,n, say, as T → ∞.
First we will prove that the family of probability measures {PN,n}
is tight for fixed n. Let η be a random variable defined on a certain
probability space (Ω̂,F , P) and uniformly distributed on [0, 1], and let,
for j = 1, ..., r,
XT,N,j,n = XT,N,j,n(σj) = fN,j,n(σj + iTη).
Then we have that
XT,N,n
def
=
(
XT,N,1,n, ..., XT,N,r,n
) D
−→
T→∞
XN,n, (12)
where XN,n is a C
r-valued random element with distribution PN,n, and
D
−→ means the convergence in distribution.
Let z1 = (z11, ..., z1r), z2 = (z21, ..., z2r) ∈ C
r. Define a metric ρ in
C
r by
A. Laurinčikas 49
ρ(z1, z2) =
( r∑
j=1
|z1j − z2j |
2
) 1
2 .
Then, clearly, this metric induces the topology of C
r.
Since the series for fn,j converges absolutely for σ > σ1j , j = 1, ..., r,
we obtain, for M > 0,
lim sup
T→∞
P
(
ρ(XT,N,n, 0) > M
)
6
6
1
M
sup
N>1
lim sup
T→∞
1
T
T∫
0
ρ
(
f
N,n
(σ1, ..., σr; t), 0
)
d t =
=
1
M
sup
N>1
lim sup
T→∞
1
T
∫ T
0
( r∑
j=1
|fN,j,n(σj + it)|2
) 1
2
d t 6
6
1
M
sup
N>1
lim sup
T→∞
(
1
T
r∑
j=1
∫ T
0
|fN,j,n(σj + it)|2 d t
) 1
2
=
=
1
M
sup
N>1
( r∑
j=1
N∑
m=1
|amj |
2v2
j (m, n)e−2λmjσj
) 1
2 6 R < ∞,
(13)
where
f
N,n
(σ1, ..., σr; t) =
(
fN,1,n(σ1 + it), ..., fN,r,n(σr + it)
)
.
Now we take M = Rǫ−1, where ǫ is an arbitrary positive number. Then
(13) yields
lim sup
T→∞
P
(
ρ(XT,N,n, 0) > M
)
6 ε.
This and (12) imply the inequality
P
(
ρ(XT,N,n, 0) > M
)
6 ε. (14)
Now we define
Kǫ = {z ∈ C
r : ρ(z, 0) 6 M}.
Then, obviously, Kǫ is a compact subset of the space C
r. In view of (14)
and of the definition of PN,n
PN,n(Kǫ) > 1 − ǫ
50 Value distribution of general Dirichlet series. VIII
for all N ∈ N. This shows that the tightness of the family {PN,n}.
Hence, by the Prokhorov theorem, see, for example, [1], the latter family
is relatively compact.
By the definition of fn,j(s) and fN,n,j(s), for σ > σ1j ,
lim
N→∞
fN,j,n(s) = fn,j(s), j = 1, ..., r,
and the series for fn,j(s) absolutely converges. Therefore, denoting
f
n
(σ1, ..., σr; t) =
(
fn,1(σ1 + it), ..., fn,r(σr + it)
)
,
we have, for every ǫ > 0 and σj > σ1j , j = 1, ..., r, that
lim
N→∞
lim sup
T→∞
ν
(
ρ(f
N,n
(σ1, ..., σr; t), fn
(σ1, ..., σr; t)) > ǫ
)
6
6 lim
N→∞
lim sup
T→∞
1
ǫT
∫ T
0
ρ(f
N,n
(σ1, ..., σr; t), fn
(σ1, ..., σr; t)) d t = 0. (15)
Define, for σj > σ1j ,
XT,j,n = XT,n(σj) = fn,j(σj + iTη), j = 1, ..., r,
and put
XT,n =
(
XT,1,n, ..., XT,r,n
)
.
Then by (15)
lim
N→∞
lim sup
T→∞
P
(
ρ(XT,N,n, XT,n) > ǫ
)
= 0. (16)
The family {PN,n} is relatively compact. Therefore, there exists a sub-
sequence {PN ′,n} ⊂ {PN,n} which converges weakly to the probability
measure Pn, say, as N ′ → ∞. Then
XN ′,n
D
−→
N ′
→∞
Pn. (17)
The space C
r is separable. Therefore, (12), (16) and (17) show that the
conditions of Theorem 4.2 from [1] are satisfied. Consequently,
XT,n
D
−→
T→∞
Pn, (18)
i.e. the measure PT,n converges weakly to the probability measure Pn on
(Cr,B(Cr)) as T → ∞.
In view of (18), the measure Pn is independent of the subsequence
{PN ′,n}. Therefore, by (17)
XN,n
D
−→
N→∞
Pn. (19)
A. Laurinčikas 51
Now, repeating the above arguments for the random elements
X̂T,N,n =
(
X̂T,N,1,n, ..., X̂T,N,r,n
)
and
X̂T,n =
(
X̂T,1,n, ..., X̂T,r,n
)
,
where
X̂T,N,j,n = X̂T,N,j,n(σj , ωj) = fN,j,n(σj + iTη, ωj), j = 1, ..., r,
X̂T,j,n = X̂T,j,n(σj , ωj) = fj,n(σj + iTη, ωj), j = 1, ..., r,
and taking into account (19), we obtain that the probability measure P̂T,n
also converges weakly to Pn as T → ∞. The theorem is proved.
5. Approximation in the mean
To pass from the functions fn,j(s) to fj(s) we need an approximation in
the mean of f1(s), ..., fr(s) and of f1(s, ω1), ..., fr(s, ωr) by fn,1(s), ..., fn,r(s)
and by fn,1(s, ω1), ..., fn,r(s, ωr), respectively. Let
f(σ1, ..., σr; t) =
(
f1(σ1 + it), ..., fr(σr + it)
)
,
and
f(σ1, ..., σr; t, ω) =
(
f1(σ1 + it, ω1), ..., fr(σr + it, ωr)
)
,
f
n
(σ1, ..., σr; t, ω) =
(
fn,1(σ1 + it, ω1), ..., fn,r(σr + it, ωr)
)
.
Theorem 7. Let σj > σ1j, j = 1, ..., r. Then
lim
N→∞
lim sup
T→∞
1
T
∫ T
0
ρ(f(σ1, ..., σr; t), fn
(σ1, ..., σr; t)) d t = 0
and
lim
N→∞
lim sup
T→∞
1
T
∫ T
0
ρ(f(σ1, ..., σr; t, ω), f
n
(σ1, ..., σr; t, ω)) d t = 0
for almost all (ω1, ..., ωr).
Proof. Suppose that the function f(s) satisfies the conditions of The-
orem A, and for σ > σ1,
fn(s) =
∞∑
m=1
amv(m, n)e−λms,
52 Value distribution of general Dirichlet series. VIII
fn(s, ω) =
∞∑
m=1
amω(m)v(m, n)e−λms,
where v(m, n) = exp{−e−(λn−λm)σ2} with σ2 > σa − σ1, and ω ∈ Ω.
Then in [12] it was obtained that, for σ > σ1,
lim
N→∞
lim sup
T→∞
1
T
∫ T
0
|f(σ + it) − fn(σ + it)|d t = 0
and
lim
N→∞
lim sup
T→∞
1
T
∫ T
0
|f(σ + it, ω) − fn(σ + it, ω)|d t = 0
for almost all ω ∈ Ω. Since
ρ(z1, z2) 6
r∑
j=1
|z1j − z2j |,
hence the theorem follows.
6. Joint limit theorems for fj(s) and fj(s, ω)
In this section we begin to prove Theorem 1. We will prove limit theorems
for the vectors f(σ1, ..., σr; t) and f(σ1, ..., σr; t, ω) defined in Section 5.
Theorem 8. Let σj > σ1j, j = 1, ..., r. Then the probability measures
PT and
P̂T (A) = νT
(
f(σ1, ..., σr; t, ω) ∈ A
)
, A ∈ B(Cr),
both converge weakly to the same probability measure on (Cr,B(Cr)) as
T → ∞.
Proof. We argue similarly to the proof of Theorem 6. By Theorem 6
the probability measures PT,n and P̂T,n converge weakly to the same
measure Pn on (Cr,B(Cr)) as T → ∞. We will show that the family of
probability measures {Pn : n ∈ N} is tight. For this, we will preserve the
notation of previous sections.
By Theorem 6
XT,n
D
−→
T→∞
Xn, (20)
where Xn is a C
r-valued random element with distribution Pn. Since the
series (11) converges and the series for each fn,j converges absolutely, we
A. Laurinčikas 53
have, for M > 0,
lim sup
T→∞
P
(
ρ(XT,n, 0) > M
)
6
6
1
M
sup
n>1
lim sup
T→∞
1
T
T∫
0
ρ
(
f
n
(σ1, ..., σr; t), 0
)
d t =
=
1
M
sup
n>1
lim sup
T→∞
1
T
∫ T
0
( r∑
j=1
|fn,j(σj + it)|2
) 1
2
d t 6
6
1
M
sup
n>1
lim sup
T→∞
(
1
T
r∑
j=1
∫ T
0
|fn,j(σj + it)|2 d t
) 1
2
=
=
1
M
sup
n>1
( r∑
j=1
∞∑
m=1
|amj |
2v2
j (m, n)e−2λmjσj
) 1
2
6
6
1
M
( r∑
j=1
∞∑
m=1
|amj |
2e−2λmjσj
) 1
2
6 R < ∞.
Hence, taking M = Rǫ−1, we find that
lim sup
T→∞
P
(
ρ(XT,n, 0) > M
)
6 ǫ.
Consequently, in view of (20)
P
(
ρ(Xn, 0) > M
)
6 ǫ.
This shows that
Pn(Kǫ) > 1 − ǫ
for all n ∈ N, i.e. the family {Pn} is tight. Hence, by the Prokhorov
theorem, it is relatively compact. Therefore, there exists a subsequence
{Pn1
} ⊂ {Pn} which converges weakly to the probability measure P , say,
on (Cr,B(Cr)) as n1 → ∞. Then
Xn1
D
−→
n1→∞
P. (21)
Let, for σj > σ1j
XT,j = XT,j(σj) = fj(σj + iTη), j = 1, ..., r,
and
XT =
(
XT,1, ..., XT,r
)
.
54 Value distribution of general Dirichlet series. VIII
Then by the first assertion of Theorem 7
lim
n→∞
lim sup
T→∞
P
(
ρ(XT,n, XT ) > ǫ
)
6
lim
n→∞
lim sup
T→∞
1
ǫT
∫ T
0
ρ
(
f
n
(σ1, ..., σr; t), f(σ1, ..., σr; t)
)
= 0.
This, (20), (21) and Theorem 4.2 of [1] show that
XT
D
−→
T→∞
P. (23)
Now let, for σj > σ1j ,
X̂T,j = X̂T,j(σj) = fj(σj + iTη, ωj), j = 1, ..., r,
and
X̂T = (X̂T,1, ..., X̂T,r).
Then, reasoning similarly as above for the vectors X̂T,n and X̂T , and
using (23) and the second assertion of Theorem 7, we obtain that the
probability measure P̂T also converges to P as T → ∞. The theorem is
proved.
7. Proof of Theorem 1
It remains to identify the limit measure P in Theorem 8. For this, we
will apply some elements of the ergodic theory.
Let at,j = {e−iλmjt : m ∈ N} for t ∈ R, j = 1, ..., r. Then, for each
j, {at,j : t ∈ R} is a one–parameter group. We define the one–parameter
family {ϕt,j : t ∈ R} of transformations on Ωj by ϕt,j = at,jωj for
ωj ∈ Ωj , j = 1, ..., r. Then we obtain a one parameter group {ϕt,j : t ∈ R}
of measurable transformations on Ωj , j = 1, ..., r.
Define {Φt : t ∈ R} = {ϕt,1 : t ∈ R} × ... × {ϕt,r : t ∈ R}. Then
{Φt : t ∈ R} is a one-parameter group of measurable transformations on
Ωr.
Lemma 9. The one-parameter group {Φt : t ∈ R} is ergodic.
Proof. In [18] it was proved that {ϕt,j : t ∈ R} for each j = 1, ..., r is
an ergodic one–parameter group. Hence the lemma follows.
Proof of Theorem 1. Let A ∈ B(Cr) be a continuity set of the measure
P in Theorem 8. Then, by Theorem 10, for σ1 > σ11, ..., σr > σ1r,
lim
T→∞
νT
(
f(σ1, ..., σr; t, ω) ∈ A
)
= P (A) (24)
A. Laurinčikas 55
for almost all ω ∈ Ωr. Now we fix the set A and define a random variable
θ on (Ωr,B(Ωr), mHr) by
θ(ω) =
{
1 if F (σ1, ..., σr; ω) ∈ A,
0 if F (σ1, ..., σr; ω) /∈ A.
Then
E(θ) =
∫
Ωr
θdmHr = mHr
(
ω ∈ Ω : F (σ1, ..., σr; ω) ∈ A
) def
= PF
is the distribution of the random element F . Since by Lemma 9 the one–
parameter group {Φt : t ∈ R} is ergodic, the random process θ(Φt(ω)) is
also ergodic. Therefore, by the Birkkhoff-Khinchine theorem
lim
T→∞
1
T
∫ T
0
θ(Φt(ω)) d t = E(θ) (26)
for almost all ω ∈ Ωr. The definitions of θ and of {Φt : t ∈ R} yield
1
T
∫ T
0
θ(Φt(ω)) d t = νT
(
f(σ1, ..., σr; t, ω) ∈ A
)
.
Hence and from (25), (26), we deduce that
lim
T→∞
νT
(
f(σ1, ..., σr; t, ω)
)
= PF (A)
for almost all ω ∈ Ωr. Consequently, by (24)
P (A) = PF (A)
for any continuity set A of the measure P . It is well known that all
continuity sets constitute the determining class. Therefore,
P (A) = PF (A)
for all A ∈ B(Cr), and the theorem is proved.
References
[1] P. Billingsley, Convergence of Probability Measures, John Wiley, New York, 1968.
[2] H. Bohr, B. Jessen, Über die Wertverteilung der Riemannschen Zetafunktion,
Erste Mitteilung, Acta Math. 54 (1930), 1-35.
[3] H. Bohr, B. Jessen, Über die Wertverteilung der Riemannschen Zetafunktion,
Zweite Mitteilung, Acta Math. 58 (1932), 1-55.
[4] J. Genys, A. Laurinčikas, Value distribution of general Dirichlet series. IV,
Lith.Math.J. 43(3)(2003), 281-294.
56 Value distribution of general Dirichlet series. VIII
[5] J. Genys, A. Laurinčikas, On joint limit theorem for general Dirichlet series,
Nonlinear Analysis: Modelling and Control, 8(2) (2003), 27-39.
[6] J. Genys, A. Laurinčikas, A joint limit theorem for general Dirichlet series,
Lith.Math.J. 44(1)(2004), 145-156.
[7] H. Heyer, Probability Measures on Locally Compact Groups, Springer–Verlag,
Berlin, Heidelberg, New York, 1977.
[8] D. Joyner, Distribution Theorems for L-functions, Longman Scientific and Tech-
nical, Harlow, 1986.
[9] A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer Aca-
demic Publishers, Dordrecht, Boston, London, 1996.
[10] A. Laurinčikas, Value-distribution of general Dirichlet series, in : Probability
Theory and Math. Statistics: Proceedings of the Seventh Vilnius Conference
(1998), B. Grigelionis et al (Eds) VSP/Utrecht, TEV/Vilnius (1999), 405-419.
[11] A. Laurinčikas, Value-distribution of general Dirichlet series. II, Lith.Math.J.
41(4)(2001), 351-360.
[12] A. Laurinčikas, Limit theorems for general Dirichlet series, Theory Stoch. Pro-
cesses, 8(24) No 3-4 (2002), 256-269.
[13] A. Laurinčikas, A joint limit theorem on the complex plane for general Dirichlet
series, Lith.Math.J. 44(3)(2004), 225-231.
[14] A. Laurinčikas, Value-distribution of general Dirichlet series. IV, Nonlinear Anal-
ysis: Modelling and Control, 10(3) (2005), 1-13.
[15] A. Laurinčikas, R. Garunkštis, The Lerch Zeta-Function, Kluwer Academic Pub-
lishers, Dordrecht, Boston, London, 2002.
[16] A. Laurinčikas, K. Matsumoto, The joint universality and the functional inde-
pendence for Lerch zeta–functions, Nagoya Math.J. 157 (2000), 211-227.
[17] A. Laurinčikas, K. Matsumoto, The joint universality of twisted automorphic
L-functions, J.Math.Soc. Japan, 56(3) (2004), 923-939.
[18] A. Laurinčikas, W. Schwarz and J. Steuding, Value distribution of general Dirich-
let series. III, in: Analytic and Probab. Methods in Number Theory, Proceedings
of the Third Intern. Conference in honour of J. Kubilius, Palanga (2001), A. Du-
bickas et al (Eds), TEV, Vilnius (2002), 137-156.
[19] A. Laurinčikas, J. Steuding, A joint limit theorem for general Dirichlet series,
Lith. Math. J. 42(2) (2002), 163-173.
[20] M. Loève, Probability Theory, Van Nostrand Company, Toronto, New York, Lon-
don 1955.
Contact information
A. Laurinčikas Vilnius University, Naugarduko 24, 03225
Vilnius, Lithuania
E-Mail: antanas.laurincikas@maf.vu.lt
Received by the editors: 11.05.2003
and in final form 06.04.2007.
|