Joint universality for $L$-functions from Selberg class and periodic Hurwitz zeta-functions
We obtain a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts consisting of $L$-functions from the Selberg class and periodic Hurwitz zeta-functions.
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| author | Macaitienė, R. Макатіене, Р. |
| author_facet | Macaitienė, R. Макатіене, Р. |
| author_sort | Macaitienė, R. |
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| description | We obtain a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts
consisting of $L$-functions from the Selberg class and periodic Hurwitz zeta-functions. |
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UDC 517.5
R. Macaitienė (Šiauliai Univ. and Šiauliai State College, Lithuania)
JOINT UNIVERSALITY FOR \bfitL -FUNCTIONS FROM SELBERG CLASS
AND PERIODIC HURWITZ ZETA-FUNCTIONS
СПIЛЬНА УНIВЕРСАЛЬНIСТЬ ДЛЯ \bfitL -ФУНКЦIЙ IЗ КЛАСУ СЕЛЬБЕРГА
ТА ПЕРIОДИЧНI ДЗЕТА-ФУНКЦIЇ ХУРВIЦА
We obtain a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts
consisting of L-functions from the Selberg class and periodic Hurwitz zeta-functions.
Встановлено теорему про спiльну унiверсальнiсть наближення сiм’ї аналiтичних функцiй сiм’єю зсувiв, що скла-
даються з L-функцiй iз класу Сельберга та перiодичних дзета-функцiй Хурвiца.
1. Introduction. Universality of zeta and L-functions is one of the most interesting phenomenons of
analytic number theory. Roughly speaking, it means that every analytic function can be approximated
with a given accuracy by shifts of the considered zeta or L-functions, uniformly on compact subsets
of a certain region. The first result in the field belongs to S. M. Voronin who discovered [20] the
universality property of the Riemann zeta-function \zeta (s), s = \sigma + it. For a modern form of the
Voronin theorem, we use the following notations. Let D =
\biggl\{
s \in \BbbC :
1
2
< \sigma < 1
\biggr\}
. Denote by \scrK the
class of compact subsets of the strip D with connected complements, by H(K), K \in \scrK , the class
of continuous functions on K which are analytic in the interior of K, and by H0(K) the subclass of
H(K) of nonvanishing on K functions. Moreover, for a measurable set A \subset \BbbR , we use the notation
measA for the Lebesgue measure of the set A. Then the following statement is well-known, for the
proof, see, for example, [7].
Theorem 1.1. Suppose that K \in \scrK and f(s) \in H0(K). Then, for every \epsilon > 0,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
T\rightarrow \infty
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\biggl\{
\tau \in [0, T ] : \mathrm{s}\mathrm{u}\mathrm{p}
s\in K
| \zeta (s+ i\tau ) - f(s)| < \epsilon
\biggr\}
> 0.
The theorem shows that a given function f(s) \in H0(K) can be approximated by shifts \zeta (s+i\tau ),
\tau \in \BbbR , from a wide set having a positive lower density.
Later, it turned out that the majority of other classical zeta and L-functions also are universal
in the above sense. A full survey on universality of zeta and L-functions is given in the excelent
paper [12]. We focus over attention on the Selberg class [17] which is one of the most extensively
studied objects of analytic number theory.
The Selberg class \scrS contains Dirichlet series
\scrL (s) =
\infty \sum
m=1
a(m)
ms
,
satisfying the following axioms:
1\circ ) analytic continuation: there exists an integer k \geq 0 such that (s - 1)k\scrL (s) is an entire
function of finite order;
c\bigcirc R. MACAITIENĖ, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5 655
656 R. MACAITIENĖ
2\circ ) Ramanujan hypothesis: a(m) \ll m\epsilon with any \epsilon > 0, where the implied constant may
depend on \epsilon ;
3\circ ) functional equation: \scrL (s) satisfies a functional equation
\Lambda \scrL (s) = w\Lambda \scrL (1 - s),
where
\Lambda \scrL (s) = \scrL (s)Qs
f\prod
j=1
\Gamma (\lambda js+ \mu j)
with positive real numbers Q, \lambda j , and complex numbers \mu j , w, \Re \mu j \geq 0 and | w| = 1;
4\circ ) Euler product: \scrL (s) has a product representation over primes p
\scrL (s) =
\prod
p
\scrL p(s),
where
\scrL p(s) = \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{ \infty \sum
k=1
b
\bigl(
pk
\bigr)
pks
\Biggr\}
with coefficients b
\bigl(
pk
\bigr)
satisfying b(pk) \ll pk\theta for some \theta <
1
2
.
Many authors investigated the structure of the class \scrS . For this, see, a survey paper [5] and
subsequent works by J. Kaczorovski and A. Perelli.
For \scrL \in \scrS , define the degree d\scrL of \scrL by
d\scrL = 2
f\sum
j=1
\lambda j ,
and let D\scrL =
\biggl\{
s \in \BbbC : \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1
2
, 1 - 1
d\scrL
\biggr)
< \sigma < 1
\biggr\}
. In [15] the analogue of Theorem 1.1
\bigl(
with
the strip of universality D\scrL
\bigr)
was obtained for \scrL \in \scrS satisfying additional hypothesis
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \infty
1
\pi (x)
\sum
p\leq x
\bigm| \bigm| a(p)\bigm| \bigm| 2 = \kappa , (1.1)
where \kappa is some positive constant (depending on \scrL ), \pi (x) =
\sum
p\leq x
1. This result is an extension of
previous result obtained by J. Steuding [18], who began to study the universality of functions from
the Selberg class.
Theorem 1.1 is an example of universality theorems for zeta and L-functions with Euler products.
The second group of universality theorems were proved for zeta-functions without Euler product.
The simplest zeta-function without Euler product is the classical Hurwitz zeta-function. Let \alpha ,
0 < \alpha \leq 1, be a fixed parameter. Then the Hurwitz zeta-function \zeta (s, \alpha ) is defined, for \sigma > 1, by
the series
\zeta (s, \alpha ) =
\infty \sum
m=0
1
(m+ \alpha )s
,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
JOINT UNIVERSALITY FOR L-FUNCTIONS FROM SELBERG CLASS . . . 657
and is meromorphically continued to the whole complex plane with unique simple pole at the point
s = 1 with residue 1. A generalization of the function \zeta (s, \alpha ) is the periodic Hurwitz zeta-function.
Let a = \{ am : m \in \BbbN 0 = \BbbN \cup \{ 0\} \} be a periodic sequence of complex numbers with minimal period
q \in \BbbN . The periodic Hurwitz zeta-function \zeta (s, \alpha ; a) is defined, for \sigma > 1, by
\zeta (s, \alpha ; a) =
\infty \sum
m=0
am
(m+ \alpha )s
.
In virtue of periodicity of the sequence a, we have that, for \sigma > 1,
\zeta (s, \alpha ; a) =
1
qs
q - 1\sum
m=0
am\zeta
\biggl(
s,
m+ \alpha
q
\biggr)
,
and the later equality gives meromorphic continuation to the whole complex plane for \zeta (s, \alpha ; a) with
a simple pole at the point s = 1. If
q - 1\sum
m=0
am = 0,
then the function \zeta (s, \alpha ; a) is entire one.
A more complicated is the joint universality of zeta-functions. In this case, a collection of analytic
functions is simultaneously approximated by a collection of shifts of zeta or L-functions. The first
joint universality theorem for Dirichlet L-functions was obtained by Voronin [21, 22]. Also, a joint
universality when analytic functions are approximated by shifts of zeta-functions, having and having
no Euler product, is possible. The first result in this direction belongs to Mishou [14]. In [6] a joint
universality theorem was proved for \zeta (s, \alpha ; a) and periodic zeta-function \zeta (s; b), which is defined,
for \sigma > 1, by
\zeta (s; b) =
\infty \sum
m=1
bm
ms
,
and b = \{ bm : m \in \BbbN \} is an another periodic sequence of complex numbers with minimal period
k \in \BbbN . The function \zeta (s; b), as \zeta (s, \alpha ; a), has meromorphic continuation to the whole complex
plane. In [8] Laurinčikas proved a generalization of the mentioned result. To state his theorem, we
need some notation. Let bj = \{ bjm : m \in \BbbN \} be a periodic sequence of complex numbers with
minimal period kj \in \BbbN and \zeta (s; bj) be the corresponding periodic zeta-function, j = 1, . . . , r1,
r1 > 1. Denote by k = [k1, . . . , kr1 ] the least common multiple of the periods k1, . . . , kr1 , by
\eta 1, . . . , \eta \varphi (k) the reduced residue system modulo k, and define the matrix
B =
\left(
b1\eta 1 b2\eta 1 . . . br1\eta 1
b1\eta 2 b2\eta 2 . . . br1\eta 2
. . . . . . . . . . . .
b1\eta \varphi (k)
b2\eta \varphi (k)
. . . br1\eta \varphi (k)
\right) ,
where \varphi (k) is the Euler function. Moreover, we suppose that, for all primes p,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
658 R. MACAITIENĖ
\infty \sum
\beta =1
| bjp\beta |
p\beta /2
\leq cj < 1, j = 1, . . . , r1. (1.2)
Further, let aj = \{ ajm : m \in \BbbN 0\} be an another periodic sequence of complex numbers with minimal
period qj \in \BbbN and \zeta (s, \alpha j ; aj) be the corresponding periodic Hurwitz zeta-function, j = 1, . . . , r2,
0 < \alpha j \leq 1. Then, in [8] the following statement was proved.
Theorem 1.2. Suppose that the sequences b1, . . . , br1 are multiplicative, inequalities (1.2) are
satisfied, the numbers \alpha 1, . . . , \alpha r2 are algebraically independent over \BbbQ , and \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(B) = r1. Let
K1, . . . ,Kr1 ,
\^K1, . . . , \^Kr2 \in \scrK , f1(s) \in H0(K1), . . . , fr1(s) \in H0(Kr1) and \^f1(s) \in H( \^K1), . . .
. . . , \^fr1(s) \in H( \^Kr2). Then, for every \varepsilon > 0,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
T\rightarrow \infty
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Biggl\{
\tau \in [0, T ] : \mathrm{s}\mathrm{u}\mathrm{p}
1\leq j\leq r1
\mathrm{s}\mathrm{u}\mathrm{p}
s\in Kj
\bigm| \bigm| \zeta (s+ i\tau ; bj) - f(s)
\bigm| \bigm| < \varepsilon ,
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq j\leq r2
\mathrm{s}\mathrm{u}\mathrm{p}
s\in \^Kj
| \zeta (s+ i\tau , \alpha j ; aj) - \^fj(s)| < \varepsilon
\Biggr\}
> 0.
The aim of this paper is a joint universality theorem for the functions \scrL (s) \in \scrS and \zeta (s, \alpha j ; ajl),
where 0 < \alpha j \leq 1 and ajl = \{ amjl : m \in \BbbN 0\} is a periodic sequence of complex numbers with
minimal period qjl \in \BbbN , j = 1, . . . , r, l = 1, . . . , lj . For j = 1, . . . , r, let qj be the least common
multiple of qj1, . . . , qjlj , and
Aj =
\left(
a1j1 a1j2 . . . a1jlj
a2j1 a2j2 . . . a2jlj
. . . . . . . . . . . .
aqjj1 aqjj2 . . . aqjjlj
\right) .
Moreover, as above, let \scrK \scrL be the class of compact subsets of the strip D\scrL with connected comple-
ments, and let H0\scrL (K), K \in \scrK \scrL , be the class of continuous non-vanishing functions on K which
are analytic in the interior of K.
Theorem 1.3. Suppose that \scrL \in \scrS , hypothesis (1.1) is satisfied, the numbers \alpha 1, . . . , \alpha r are
algebraically independent over \BbbQ , and that \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(Aj) = lj , j = 1, . . . , r. Let K \in \scrK \scrL , f(s) \in
\in H0\scrL (K), and, for every j = 1, . . . , r and l = 1, . . . , lj , let Kjl \in \scrK and fjl(s) \in H(Kjl). Then,
for every \varepsilon > 0,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
T\rightarrow \infty
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Biggl\{
\tau \in [0, T ] : \mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| \scrL (s+ i\tau ) - f(s)
\bigm| \bigm| < \varepsilon ,
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq j\leq r
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq l\leq lj
\mathrm{s}\mathrm{u}\mathrm{p}
s\in Kjl
\bigm| \bigm| \zeta (s+ i\tau , \alpha j ; ajl) - fjl(s)
\bigm| \bigm| < \varepsilon
\Biggr\}
> 0.
Clearly, Theorem 1.3 implies the results of previous works [4, 10, 11, 16], where instead \scrL \in \scrS ,
the function \zeta (s), zeta-functions of normalized Hecke cusp forms, zeta-functions of newforms, and
zeta-functions of new forms with Dirichlet character were taken, respectively. For example, we can
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
JOINT UNIVERSALITY FOR L-FUNCTIONS FROM SELBERG CLASS . . . 659
take, for r = 2,
\alpha 1 = 2 -
3\surd 2, \alpha 2 = 2 -
3\surd 4,
because the numbers \alpha 1 = 2
3\surd 2 and \alpha 2 = 2
3\surd 4 are algebraically independent over \BbbQ [2].
2. Limit theorem. For the proof of Theorem 1.3, we will prove a limit theorem on weakly
convergent probability measures in the space of analytic functions. Let G be a region on the complex
plane. Denote by H(G) the space of analytic functions on G equipped with the topology of uniform
convergence on compacta. Let
u =
r\sum
j=1
lj , v = u+ 1,
and
Hv = Hv(D\scrL , D) = H(D\scrL )\times Hu(D).
As usual, denote by \scrB (X) the Borel \sigma -field of the space X. For brevity, let \alpha = (\alpha 1, . . . , \alpha r),
a = (a11, . . . , a1l1 , . . . , ar1, . . . , arlr) and
Z(s1, s, \alpha ; a,\scrL ) =
\bigl(
\scrL (s1), \zeta (s, \alpha 1; a11), . . . , \zeta (s, \alpha 1; a1l1), . . . , \zeta (s, \alpha r; ar1), . . . , \zeta (s, \alpha r; arlr)
\bigr)
.
In this section, we consider the weak convergence for
PT (A)
df
=
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
\tau \in [0, T ] : Z(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL ) \in A
\bigr\}
, A \in \scrB (Hv),
as T \rightarrow \infty . To state a limit theorem for PT , we need a certain topological structure. Let
\gamma = \{ s \in \BbbC : | s| = 1\} . Define
\^\Omega =
\prod
p
\gamma p and \Omega =
\prod
m\in \BbbN 0
\gamma m,
where \gamma p = \gamma for all primes p and \gamma m = \gamma for all m \in \BbbN 0. By the Tikhonov theorem, the tori
\^\Omega and \Omega with the product topologies and pointwise multiplication are compact topological groups.
Thus, on (\^\Omega ,\scrB (\^\Omega )) and
\bigl(
\Omega ,\scrB (\Omega )
\bigr)
the probability Haar measures \^mH and mH , respectively, exist,
and we have the probability spaces (\^\Omega ,\scrB (\^\Omega ), \^mH),
\bigl(
\Omega ,\scrB (\Omega ),mH
\bigr)
. Moreover, for j = 1, . . . , r, let
\Omega j = \Omega , and
\Omega = \^\Omega \times \Omega 1 \times . . .\times \Omega r.
Then again, \Omega is a compact topological Abelian group, and this gives one more probability space
(\Omega ,\scrB (\Omega ),mH), where mH is the probability Haar measure on (\Omega ,\scrB (\Omega )). Denote by \^\omega (p) the
projection of \^\omega \in \^\Omega to \gamma p, p \in \scrP (\scrP is the set of all prime numbers), and by \omega j(m) the projection
of \omega j \in \Omega j to \gamma m, m \in \BbbN 0. After the above notation, on the probability space (\Omega ,\scrB (\Omega ),mH) define
the Hv -valued random element Z(s1, s, \omega , \alpha ; a,\scrL ), \omega = (\^\omega , \omega 1, . . . , \omega r) \in \Omega , by the formula
Z(s1, s, \omega , \alpha ; a,\scrL ) =
\bigl(
\scrL (s1, \^\omega ), \zeta (s, \alpha 1, \omega 1; a11), . . . , \zeta (s, \alpha 1, \omega 1; a1l1), . . .
. . . , \zeta (s, \omega r, \alpha r; ar1), . . . , \zeta (s, \omega r, \alpha r; arlr)
\bigr)
,
where, for s1 \in D\scrL ,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
660 R. MACAITIENĖ
\scrL (s1, \^\omega ) =
\infty \sum
m=1
a(m)\^\omega (m)
ms1
with
\^\omega (m) =
\prod
pl| m
pl+1\nmid m
\^\omega l(p), m \in \BbbN ,
and, for s \in D,
\zeta (s, \alpha j , \omega j ; ajl) =
\infty \sum
m=0
amjl\omega j(m)
(m+ \alpha j)s
, j = 1, . . . , r, l = 1, . . . , lj .
We note that, for almost all \^\omega \in \^\Omega , \scrL (s1, \^\omega ) has the Euler product [15], i.e.,
\scrL (s1, \^\omega ) = \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl\{ \sum
p
\infty \sum
k=1
b(pk)\^\omega k(p)
pks1
\Biggr\}
.
Denote by PZ the distribution of the random element Z(s1, s, \omega , \alpha ; a,\scrL ), i.e.,
PZ(A) = mH
\bigl(
\omega \in \Omega : Z(s1, s, \omega , \alpha ; a,\scrL ) \in A
\bigr)
, A \in \scrB (Hv).
Theorem 2.1. Suppose that \scrL \in \scrS , hypothesis (1.1) is satisfied, and the numbers \alpha 1, . . . , \alpha r
are algebraically independent over \BbbQ . Then PT converges weakly to PZ as T \rightarrow \infty .
A way of the proof is sufficiently well-known, see, for example, similar theorems from [4, 10, 11],
therefore, we will present only principal moments of the proof.
Lemma 2.1. Suppose that the numbers \alpha 1, . . . , \alpha r are algebraically independent over \BbbQ . Then
QT (A)
df
=
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Bigl\{
\tau \in [0, T ] :
\bigl(
(p - i\tau : p \in \scrP ),
\bigl(
(m+ \alpha 1)
- i\tau : m \in \BbbN 0
\bigr)
, . . . ,
\bigl(
(m+ \alpha r)
- i\tau : m \in \BbbN 0)
\bigr)
\in A
\Bigr\}
, A \in \scrB (\Omega ),
converges weakly to the Haar measure mH as T \rightarrow \infty .
Proof of the lemma is given in [8] (Theorem 3). However, for conveniences of a reader, we
rewrite the proof. It is well known that the dual group of \Omega is isomorphic to the group
G =
\left( \bigoplus
p\in \scrP
\BbbZ p
\right) r\bigoplus
j=1
\left( \bigoplus
m\in \BbbN 0
\BbbZ jm
\right) ,
where \BbbZ p = \BbbZ for all p \in \scrP and \BbbZ jm = \BbbZ for all m \in \BbbN 0 and j = 1, . . . , r. An element
(k, l1, . . . , lr) =
\bigl(
(kp : p \in \scrP ), (l1m : m \in \BbbN 0), . . . , (lrm : m \in \BbbN 0)
\bigr)
of the group G, where only a
finite number of integers kp and l1m, . . . , lrm are distinct from zero, acts on \Omega by
(\^\omega , \omega 1, . . . , \omega r) \rightarrow (\^\omega k, \omega l1
1 , . . . , \omega
lr
r ) =
\prod
p\in \scrP
\^\omega kp(p)
r\prod
j=1
\prod
m\in \BbbN 0
\omega
ljm
j (m).
Therefore, the right-hand side of the latter equality defines characters of \Omega , hence, the Fourier
transform gT (k, l1, . . . , lr) of QT is of the form
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JOINT UNIVERSALITY FOR L-FUNCTIONS FROM SELBERG CLASS . . . 661
gT (k, l1, . . . , lr) =
\int
\Omega
\left( \prod
p\in \scrP
\^\omega kp(p)
r\prod
j=1
\prod
m\in \BbbN 0
\omega
ljm
j (m)
\right) dQT .
Thus, by the definition of QT ,
gT (k, l1, . . . , lr) =
1
T
T\int
0
\left( \prod
p\in \scrP
p - i\tau kp
r\prod
j=1
\prod
m\in \BbbN 0
(m+ \alpha j)
- i\tau ljm
\right) d\tau =
=
1
T
T\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\left\{ - i\tau
\left( \sum
p\in \scrP
kp \mathrm{l}\mathrm{o}\mathrm{g} p+
r\sum
j=1
\sum
m\in \BbbN 0
ljm \mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha j)
\right) \right\} d\tau , (2.1)
where, as above, only a finite number of integers kp and l1m, . . . , lrm are distinct from zero. Clearly,
gT (0, 0, . . . , 0) = 1. (2.2)
Now suppose that (k, l1, . . . , lr) \not = (0, 0, . . . , 0). Since the numbers \alpha 1, . . . , \alpha r are algebraically
independent over \BbbQ , the set
\bigl\{
(\mathrm{l}\mathrm{o}\mathrm{g}(m + \alpha 1) : m \in \BbbN 0), . . . , (\mathrm{l}\mathrm{o}\mathrm{g}(m + \alpha r) : m \in \BbbN 0)
\bigr\}
is linearly
independent over \BbbQ . Indeed, suppose that there exist l11, . . . , l1k1 , . . . , lr1, . . . , lrkr \in \BbbZ \setminus \{ 0\} and
m11, . . . ,mrk1 , . . .mr1, . . . ,mrkr such that
l11 \mathrm{l}\mathrm{o}\mathrm{g}(m11 + \alpha 1) + . . .+ l1k1 \mathrm{l}\mathrm{o}\mathrm{g}(m1k1 + \alpha 1) + . . .
. . .+ lr1 \mathrm{l}\mathrm{o}\mathrm{g}(mr1 + \alpha r) + . . .+ lrkr \mathrm{l}\mathrm{o}\mathrm{g}(mrkr + \alpha r) = 0.
Then
(m11 + \alpha 1)
l11 . . . (m1k1 + \alpha 1)
l1k1 . . . (mr1 + \alpha r)
lr1 . . . (mrkr + \alpha r)
lrkr = 1.
From this, it follows that there exists a polynomial p(x1, . . . , xr) with integer coefficients such that
p(\alpha 1, . . . , \alpha r) = 0, and this contradicts the algebraic independence of the numbers \alpha 1, . . . , \alpha r. It is
well known that the set \{ \mathrm{l}\mathrm{o}\mathrm{g} p : p \in \scrP \} is linearly independent over \BbbQ . Thus, assuming that the set\bigl\{
(\mathrm{l}\mathrm{o}\mathrm{g} p : p \in \scrP ), (\mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha 1) : m \in \BbbN 0), . . . , (\mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha r) : m \in \BbbN 0)
\bigr\}
(2.3)
is linearly dependent over \BbbQ , we obtain, similarly as in the case of the set\bigl\{
(\mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha 1) : m \in \BbbN 0), . . . , (\mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha r) : m \in \BbbN 0)
\bigr\}
,
the contradiction to algebraic independence of \alpha 1, . . . , \alpha r. Since the sums in (2.1) are finite, from
the linear independence of (2.3), we find that, in the case (k, l1, . . . , lr) \not = (0, 0, . . . , 0),
\sum
p\in \scrP
kp \mathrm{l}\mathrm{o}\mathrm{g} p+
r\sum
j=1
\sum
m\in \BbbN 0
ljm \mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha j) \not = 0.
After integration, we find from (2.1) and (2.2)
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662 R. MACAITIENĖ
gT (k, l1, . . . , lr) =
=
\left\{
1, if (k, l1, . . . , lr) = (0, 0, . . . , 0),
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\Bigl\{
- iT
\Bigl( \sum
p\in \scrP
kp \mathrm{l}\mathrm{o}\mathrm{g} p+
\sum r
j=1
\sum
m\in \BbbN 0
ljm \mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha j)
\Bigr) \Bigr\}
iT
\Bigl( \sum
p\in \scrP
kp \mathrm{l}\mathrm{o}\mathrm{g} p+
\sum r
j=1
\sum
m\in \BbbN 0
ljm \mathrm{l}\mathrm{o}\mathrm{g}(m+ \alpha j)
\Bigr) ,
if (k, l1, . . . , lr) \not = (0, 0, . . . , 0).
Hence,
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
gT (k, l1, . . . , lr) =
\left\{ 1, if (k, l1, . . . , lr) = (0, 0, . . . , 0),
0, if (k, l1, . . . , lr) \not = (0, 0, . . . , 0).
(2.4)
It is not difficult to see, that the right-hand side of (2.4) is the Fourier transform of the Haar measure
mH . Really, mH is the product of Haar measures on each circle of \Omega . The Haar measure on the
unit circle coincides with the Lebesgue measure. Therefore, for example, denoting by \mu p the Haar
measure on \gamma p, we find that
\int
\gamma p
\^\omega kp(p)d\mu p =
1\int
0
e2\pi ikpxdx =
\left\{ 1, if kp = 0,
0, if kp \not = 0.
Now, the assertion of the theorem follows from general continuity theorems for probability mea-
sures on compact groups (if the Fourier transform converge, then the corresponding measure con-
verges weakly to the measure of the limit of Fourier transform), see, for example [3] (Theorem 1.4.2).
Now, for fixed \sigma 1 >
1
2
, let
un(m) = \mathrm{e}\mathrm{x}\mathrm{p}
\Bigl\{
-
\Bigl( m
n
\Bigr) \sigma 1
\Bigr\}
, m, n \in \BbbN ,
and
un(m,\alpha j) = \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
-
\biggl(
m+ \alpha j
n+ \alpha j
\biggr) \sigma 1
\biggr\}
, m \in \BbbN 0, n \in \BbbN , j = 1, . . . , r,
and define
\scrL n(s) =
\infty \sum
m=1
a(m)un(m)
ms
and
\zeta n(s, \alpha j ; ajl) =
\infty \sum
m=0
amjlun(m,\alpha j)
(m+ \alpha j)s
, j = 1, . . . , r, l = 1, . . . , lj .
Then we have that the series for \scrL n(s) converges absolutely for \sigma > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1
2
, 1 - 1
d\scrL
\biggr)
[19], and
the series for \zeta n(s, \alpha j ; ajl) converges absolutely for \sigma >
1
2
. Also, we define
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\scrL n(s, \^\omega ) =
\infty \sum
m=1
a(m)\^\omega (m)un(m)
ms
(2.5)
and
\zeta n(s, \omega j , \alpha j ; ajl) =
\infty \sum
m=0
amjl\omega j(m)un(m,\alpha j)
(m+ \alpha j)s
, j = 1, . . . , r, l = 1, . . . , lj . (2.6)
In the next step of the proof, consider the weak convergence of the measures
PT,n(A) =
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
\tau \in [0, T ] : Zn(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL ) \in A
\bigr\}
, A \in \scrB (Hv),
and
\^PT,n(A) =
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
\tau \in [0, T ] : Zn(s1 + i\tau , s+ i\tau , \omega , \alpha ; a,\scrL ) \in A
\bigr\}
, A \in \scrB (Hv),
as T \rightarrow \infty , where
Zn(s1, s, \alpha ; a,\scrL ) =
\bigl(
\scrL n(s1), \zeta n(s, \alpha 1; a11), . . . , \zeta n(s, \alpha 1; a1l1), . . .
. . . , \zeta n(s, \alpha r; ar1), . . . , \zeta n(s, \alpha r; arlr)
\bigr)
and
Zn(s1, s, \omega , \alpha ; a,\scrL ) =
\bigl(
\scrL n(s1, \^\omega ), \zeta n(s, \omega 1, \alpha 1; a11), . . . , \zeta n(s, \omega 1, \alpha 1; a1l1), . . .
. . . , \zeta n(s, \omega r, \alpha r; ar1), . . . , \zeta n(s, \omega r, \alpha r; arlr)
\bigr)
.
Lemma 2.2. Suppose that the numbers \alpha 1, . . . , \alpha r are algebraically independent over \BbbQ . Then,
for every fixed \omega \in \Omega , the measures PT,n and \^PT,n both converge weakly to the same probability
measure Pn on
\bigl(
Hv,\scrB (Hv)
\bigr)
as T \rightarrow \infty .
Proof. We apply a standard method based on the preservation of weak convergence under con-
tinuous mapping, and Theorem 5.1 of [1]. Define the function hn : \Omega \rightarrow Hv by the formula
hn(\omega ) = Zn(s1, s, \omega , \alpha ; a,\scrL ).
The absolute convergence of the series (2.5) and (2.6) shows that the function hn is continuous.
Moreover,
hn
\bigl( \bigl(
p - i\tau : p \in \scrP
\bigr)
,
\bigl(
(m+ \alpha 1)
- i\tau : m \in \BbbN 0
\bigr)
, . . . ,
\bigl(
(m+ \alpha r)
- i\tau : m \in \BbbN 0
\bigr) \bigr)
=
= Zn(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL ).
Hence, PT,n = QTh
- 1
n . This, the continuity of hn, Lemma 2.1 and Theorem 5.1 of [1] imply the
weak convergence of the measure PT,n to mHh - 1
n as T \rightarrow \infty .
Now, for a fixed \omega 0 \in \Omega , let h(\omega ) = \omega \omega 0 for \omega \in \Omega . Then, obviously,
hn
\bigl(
h
\bigl(
(p - i\tau : p \in \scrP ),
\bigl(
(m+ \alpha 1)
- i\tau : m \in \BbbN 0
\bigr)
, . . . ,
\bigl(
(m+ \alpha r)
- i\tau : m \in \BbbN 0
\bigr) \bigr) \bigr)
=
= Zn(s1 + i\tau , s+ i\tau , \omega 0, \alpha ; a,\scrL ).
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664 R. MACAITIENĖ
Therefore, repeating the above arguments and using the invariance of the Haar measure mH , we
find that the measure \^PT,n also converges weakly to mHh - 1
n as T \rightarrow \infty . Thus, the measures PT,n
and \^PT,n both converge weakly to the measure Pn = mHh - 1
n as T \rightarrow \infty .
The change Zn by Z requires certain approximation results. Denote by \rho \scrL , \rho and \rho v the metrices
on H(D\scrL ), H(D) and Hv, respectively, inducing the topology of uniform convergence on compacta.
We note that, for g = (g, g11, . . . , g1l1 , . . . , gr1, . . . , grlr), f = (f, f11, . . . , f1l1 , . . . , fr1, . . . , frlr) \in
\in Hv,
\rho v(g, f) = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
\rho \scrL (g, f), \mathrm{m}\mathrm{a}\mathrm{x}
1\leq j\leq r
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq l\leq lj
\rho (gjl, fjl)
\biggr)
.
Lemma 2.3. We have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\rho v
\bigl(
Z(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL ), Zn(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL )
\bigr)
d\tau = 0.
Suppose that the numbers \alpha 1, . . . , \alpha r are algebraically independent over \BbbQ . Then, for almost all
\omega \in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\rho v
\bigl(
Z(s1 + i\tau , s+ i\tau , \omega , \alpha ; a,\scrL ), Zn(s1 + i\tau , s+ i\tau , \omega , \alpha ; a,\scrL )
\bigr)
d\tau = 0.
Proof. By the definition of the metric \rho v,
1
T
T\int
0
\rho v
\bigl(
Z(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL ), Zn(s1 + i\tau , s+ i\tau , \alpha ; a,\scrL )
\bigr)
d\tau \leq
\leq 1
T
T\int
0
\rho \scrL
\bigl(
\scrL (s1 + i\tau ),\scrL n(s1 + i\tau )
\bigr)
d\tau +
+
r\sum
j=1
lj\sum
l=1
1
T
T\int
0
\rho
\bigl(
\zeta (s+ i\tau , \alpha j ; ajl), \zeta n(s+ i\tau , \alpha j ; ajl)
\bigr)
d\tau . (2.7)
Moreover, in view of Lemma 4.8 from [19],
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\rho \scrL
\bigl(
\scrL (s1 + i\tau ),\scrL n(s1 + i\tau )
\bigr)
d\tau = 0, (2.8)
and, by the proof of Lemma 2.4 from [11],
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\rho
\bigl(
\zeta (s+ i\tau , \alpha j ; ajl), \zeta n(s+ i\tau , \alpha j ; ajl)
\bigr)
d\tau = 0 (2.9)
for j = 1, . . . , r, l = 1, . . . , lj . Therefore, the first assertion of the lemma follows from (2.7) – (2.9).
Similarly, in view of Lemma 4.10 from [19], for almost all \^\omega with respect to the measure \^mH ,
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\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\rho \scrL
\bigl(
\scrL (s1 + i\tau , \^\omega ),\scrL n(s1 + i\tau , \^\omega )
\bigr)
d\tau = 0. (2.10)
Let
\rho u(g, f) = \mathrm{m}\mathrm{a}\mathrm{x}
1\leq j\leq r
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq l\leq lj
\rho (gjl, fjl)
and let \widetilde mH denote the Haar measure on
\bigl(
\Omega 1 \times . . . \times \Omega r,\scrB (\Omega 1 \times . . . \times \Omega r)
\bigr)
. Then formula (2.6)
of [4] asserts that, for almost all (\omega 1, . . . , \omega r) \in \Omega 1 \times . . .\times \Omega r with respect to \widetilde mH ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\rho u
\bigl( \bigl(
\zeta (s+ i\tau , \omega 1, \alpha 1; a11), . . . , \zeta (s+ i\tau , \omega r, \alpha r; arlr)
\bigr)
,
\bigl(
\zeta n(s+ i\tau , \omega 1, \alpha 1; a11), . . . , \zeta n(s+ i\tau , \omega r, \alpha r; arlr)
\bigr) \bigr)
d\tau = 0. (2.11)
The measure mH is the product of the measures \^mH and \widetilde mH . Therefore, the second assertion of
the lemma follows from (2.10), (2.11) and the analogue of equality (2.7).
Lemma 2.4. Suppose that \scrL \in \scrS , hypothesis (1.1) is satisfied and the numbers \alpha 1, . . . , \alpha r are
algebraically independent over \BbbQ . Then the probability measures PT and
\^PT (A) =
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
\tau \in [0, T ] : Z(s1 + i\tau , s+ i\tau , \omega , \alpha ; a,\scrL ) \in A
\bigr\}
, A \in \scrB (Hv),
both converge weakly, for almost all \omega \in \Omega , to the same probability measure P on
\bigl(
Hv,\scrB (Hv)
\bigr)
as
T \rightarrow \infty .
Proof. The properties of the class \scrS implies that, for \sigma >
1
2
,
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
T
T\int
0
\bigm| \bigm| \scrL n(\sigma + it)
\bigm| \bigm| 2dt = \infty \sum
m=1
| a(m)| u2n(m)
m2\sigma
\leq
\infty \sum
m=1
| a(m)| 2
m2\sigma
< \infty .
This and the Cauchy integral formula lead, for a compact set K of D\scrL , to the estimate
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| \scrL n(s1 + i\tau )
\bigm| \bigm| d\tau \leq CK
\Biggl( \infty \sum
m=1
| a(m)| 2
m2\sigma K
\Biggr) 1/2
(2.12)
with some CK and \sigma K > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1
2
, 1 - 1
d\scrL
\biggr)
. By (2.5) of [11], for a compact subset K of D,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
T\int
0
\mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| \zeta n(s+ i\tau , \alpha j ; ajl)
\bigm| \bigm| d\tau \leq BK
\Biggl( \infty \sum
m=0
| amjl| 2
(m+ \alpha )2\^\sigma K
\Biggr) 1/2
(2.13)
with some BK > 0 and \^\sigma K >
1
2
for all j = 1, . . . , r, l = 1, . . . , lj .
Let \theta be a random variable on a certain probability space
\bigl( \widetilde \Omega ,\scrA ,\BbbP
\bigr)
and uniformly distributed on
[0, 1]. On this probability space define the Hv -valued random element XT,n by the formula
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666 R. MACAITIENĖ
XT,n = XT,n(s1, s) =
=
\bigl(
XT,n(s1), XT,n,1,1(s), . . . , XT,n,1,l1(s), . . . , XT,n,r,1(s), . . . , XT,n,r,lr(s)
\bigr)
=
= Zn (s1 + i\theta T, s+ i\theta T, \alpha ; a,\scrL ) .
Then, by Lemma 2.2, we have that
XT,n
\scrD - \rightarrow Xn, (2.14)
where
\scrD - \rightarrow means the convergence in distribution, and
Xn = Xn(s1, s) =
\bigl(
Xn(s1), Xn,1,1(s), . . . , Xn,1,l1(s), . . . , Xn,r,1(s), . . . , Xn,r,lr(s)
\bigr)
is the Hv -valued random element with the distribution Pn (Pn is the limit measure in Lemma 2.2).
Using (2.12) – (2.14), we prove, in a standard way, see, for example, [11, 19], that the family of
probability measures \{ Pn : n \in \BbbN \} is tight, i.e., that, for every \epsilon > 0, there exists a compact set
K = K(\epsilon ) \subset Hv such that Pn(K) > 1 - \epsilon for all n \in \BbbN . Indeed, let the compact sets \^Km and Km
come from the definition of the metric \rho v (the definition of the metric \rho is given in [8]). Let
\^Rm = C \^Km
\Biggl( \infty \sum
m=1
| a(m)| 2)
k2\sigma \^Km
\Biggr) 1/2
and
Rjlm = BKm
\Biggl( \infty \sum
m=0
| amjl| 2
(m+ \alpha j)2\sigma Km
\Biggr) 1/2
.
Let \epsilon > 0 be an arbitrary number, and
\^Mm = \^Rm2m+1\epsilon - 1, Mjlm = Rjlm2u+m+1\epsilon - 1, m \in \BbbN .
Then, in virtue of (2.12) and (2.13),
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
\BbbP
\Biggl( \Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s1\in \^Km
\bigm| \bigm| XT,n(s1)
\bigm| \bigm| > \^Mm
\Biggr)
or
\biggl(
\exists j, l : \mathrm{s}\mathrm{u}\mathrm{p}
s\in Km
\bigm| \bigm| XT,n,j,l(s)
\bigm| \bigm| > Mjlm
\biggr) \Biggr)
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
\BbbP
\Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s1\in \^Km
\bigm| \bigm| XT,n(s1)
\bigm| \bigm| > \^Mm
\Biggr)
+
+ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
\BbbP
\biggl(
\exists j, l : \mathrm{s}\mathrm{u}\mathrm{p}
s\in Km
\bigm| \bigm| XT,n,j,l(s)
\bigm| \bigm| > Mjlm
\biggr)
\leq
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Biggl\{
\tau \in [0, T ] : \mathrm{s}\mathrm{u}\mathrm{p}
s1\in \^Km
\bigm| \bigm| \scrL (s1 + i\tau )
\bigm| \bigm| > \^Mm
\Biggr\}
+
+ \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
r\sum
j=1
lj\sum
l=1
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\biggl\{
\tau \in [0, T ] : \mathrm{s}\mathrm{u}\mathrm{p}
s\in Km
\bigm| \bigm| \zeta (s+ i\tau , \alpha j ; ajl)
\bigm| \bigm| > Mjlm
\biggr\}
\leq
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JOINT UNIVERSALITY FOR L-FUNCTIONS FROM SELBERG CLASS . . . 667
\leq \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
T \^Mm
T\int
0
\mathrm{s}\mathrm{u}\mathrm{p}
s1\in \^Km
| \scrL (s1 + i\tau )| d\tau +
+
r\sum
j=1
lj\sum
l=1
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
1
TMjlm
T\int
0
\mathrm{s}\mathrm{u}\mathrm{p}
s\in Km
\bigm| \bigm| \zeta (s+ i\tau , \alpha j ; ajl)
\bigm| \bigm| d\tau \leq \epsilon
2m
.
Hence, by (2.14),
\BbbP
\Biggl( \Biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
s1\in \^Km
| Xn(s1)| > \^Mm
\Biggr)
or
\biggl(
\exists j, l : \mathrm{s}\mathrm{u}\mathrm{p}
s\in Km
\bigm| \bigm| Xn,j,l(s)
\bigm| \bigm| > Mjlm
\biggr) \Biggr)
\leq \epsilon
2m
, (2.15)
j = 1, . . . , r, l = 1, . . . , lj . Define the set
Kv
\epsilon =
\Bigl\{
f \in Hv : \mathrm{s}\mathrm{u}\mathrm{p}
s1\in \^Km
\bigm| \bigm| f(s1)\bigm| \bigm| \leq \^Mm, \mathrm{s}\mathrm{u}\mathrm{p}
s\in Km
\bigm| \bigm| fjl(s)\bigm| \bigm| \leq Mjlm, j = 1, . . . , r, l = 1, . . . , lj , m \in \BbbN
\Bigr\}
.
Then Kv
\epsilon is a compact set in Hv. Moreover, by (2.15),
Pn
\bigl(
Kv
\epsilon
\bigr)
\geq 1 - \epsilon
for all n \in \BbbN , i.e., \{ Pn : n \in \BbbN \} is tight. Hence, by the Prokhorov theorem [1] (Theorem 6.1).
(If the family of probability measures is tight, then it is relatively compact), this family is relatively
compact. Therefore, there exists a sequence Xnk
and a probability measure P on
\bigl(
Hv,\scrB (Hv)
\bigr)
such that
Xnk
\scrD - \rightarrow
k\rightarrow \infty
P. (2.16)
On (\widetilde \Omega ,\scrA ,\BbbP ), define one more Hv -valued random element XT by the formula
XT = XT (s1, s) = Z
\bigl(
s1 + i\theta T, s+ i\theta T, \alpha ; a,\scrL )
\bigr)
.
Then, in view of Lemma 2.3, we obtain that, for every \epsilon > 0,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
T\rightarrow \infty
\BbbP
\bigl(
\rho v(XT , XT,n) \geq \epsilon
\bigr)
= 0.
This, (2.14) and (2.16) show that all hypothesis of Theorem 4.2 of [1] are satisfied. Therefore, by
this theorem
XT
\scrD - \rightarrow
T\rightarrow \infty
P,
which is equivalent to the weak convergence of PT to P as T \rightarrow \infty . From this, it follows that the
measure P is independent on the sequence Xnk
. Thus,
Xn
\scrD - \rightarrow
n\rightarrow \infty
P. (2.17)
Similarly, using the Hv -valued random elements
Zn
\bigl(
s1 + i\theta T, s+ i\theta T, \omega , \alpha ; a,\scrL
\bigr)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
668 R. MACAITIENĖ
and
Z
\bigl(
s1 + i\theta T, s+ i\theta T, \omega , \alpha ; a,\scrL
\bigr)
as well as relation (2.17), we obtain that the measure \^PT also converges weakly to P as T \rightarrow \infty .
Proof of Theorem 2.1. In virtue of Lemma 2.4, it suffices to show that P = PZ . Let A be a
continuity set of the measure P. On the probability space
\bigl(
\Omega ,\scrB (\Omega ),mH
\bigr)
, define the random variable
\xi by the formula
\xi (\omega ) =
\left\{ 1, if Z(s1, s, \omega , \alpha ; a,\scrL ) \in A,
0, otherwise.
Lemma 2.4 implies the relation
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
\^PT (A) = P (A). (2.18)
By the definition of \xi , the expectation \BbbE \xi is
\BbbE \xi =
\int
\Omega
\xi dmH = mH
\bigl(
\omega \in \Omega : Z(s1, s, \omega , \alpha ; a,\scrL ) \in A
\bigr)
= PZ(A). (2.19)
For \tau \in \BbbR , define the transformation \Phi \tau of \Omega by
\Phi \tau (\omega ) =
\Bigl( \bigl(
p - i\tau : p \in \scrP
\bigr)
,
\bigl(
(m+ \alpha 1)
- i\tau : m \in \BbbN 0
\bigr)
, . . . ,
\bigl(
(m+ \alpha r)
- i\tau : m \in \BbbN 0
\bigr) \Bigr)
\omega ,
\omega \in \Omega . Lemma 7 of [8] asserts that the group of measurable measure preserving transformations
\{ \Phi \tau : \tau \in \BbbR \} is ergodic. Hence, the random process \xi
\bigl(
\Phi \tau (\omega )
\bigr)
is ergodic as well. Therefore, by the
classical Birkhoff – Khintchine theorem,
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
T
T\int
0
\xi
\bigl(
\Phi \tau (\omega )
\bigr)
d\tau = \BbbE \xi . (2.20)
On the other hand, by the definitions of \xi and \Phi \tau ,
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
T
T\int
0
\xi
\bigl(
\Phi \tau (\omega )
\bigr)
d\tau =
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
\tau \in [0, T ] : Z(s1 + i\tau , s+ i\tau , \omega , \alpha ; a,\scrL ) \in A
\bigr\}
.
This, (2.19) and (2.20) show that
\mathrm{l}\mathrm{i}\mathrm{m}
T\rightarrow \infty
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\bigl\{
\tau \in [0, T ] : Z(s1 + i\tau , s+ i\tau , \omega , \alpha ; a,\scrL ) \in A
\bigr\}
= PZ(A).
Therefore, in view of (2.18) and the definition of \^PT , P (A) = PZ(A) for all continuity sets A of
the measure P. Since all continuity sets constitute a determining class, we have that P = PZ .
The theorem is proved.
3. Support. We recall that the support of PZ is the minimal closed set SPZ
\subset Hv such that
PZ(SPZ
) = 1. The set SPZ
consists of all points g \in Hv such that, for every open neighbourhood G
of g, the inequality PZ(G) > 0 is satisfied.
Let
S\scrL =
\bigl\{
g \in H(D\scrL ) : g(s) \not = 0 or g(s) \equiv 0
\bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
JOINT UNIVERSALITY FOR L-FUNCTIONS FROM SELBERG CLASS . . . 669
Theorem 3.1. Suppose that \scrL \in \scrS , hypothesis (1.1) is satisfied, the numbers \alpha 1, . . . , \alpha r are
algebraically independent over \BbbQ , and that \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(Aj) = lj , j = 1, . . . , r. Then the support of the
measure PZ is the set S\scrL \times Hu(D).
Proof. We have that
Hv = H(D\scrL )\times Hu(D). (3.1)
Since the spaces H(D\scrL ) and Hu(D) are separable, by (3.1),
\scrB (Hv) = \scrB
\bigl(
H(D\scrL )
\bigr)
\times \scrB
\bigl(
Hu(D)
\bigr)
.
Therefore, it suffices to consider PZ(A) for A = A1 \times A2 with A1 \in H(D\scrL ) and A2 \in Hu(D).
The measure mH is the product of the measures \^mH and \widetilde mH . Hence,
PZ(A) = mH
\bigl(
\omega \in \Omega : Z(s1, s, \omega , \alpha ; a,\scrL ) \in A
\bigr)
=
= mH
\Bigl(
\omega \in \Omega : \scrL (s1, \^\omega ) \in A1,
\bigl(
\zeta (s, \alpha 1, \omega 1; a11), . . . , \zeta (s, \alpha r, \omega r; arlr)
\bigr)
\in A2
\Bigr)
=
= \^mH
\bigl(
\^\omega \in \^\Omega : \scrL (s1, \^\omega ) \in A1
\bigr) \widetilde mH
\Bigl(
(\omega 1, . . . , \omega r) \in \Omega 1 \times . . .\times \Omega r :\Bigl(
\zeta (s, \alpha 1, \omega 1; a11), . . . , \zeta (s, \alpha r, \omega r; arlr)
\Bigr)
\in A2
\Bigr)
. (3.2)
By Proposition 3 of [15], the support of the random element \scrL (s1, \^\omega ) is the set S\scrL . We note that
in [15], the H(D\scrL ,N )-valued random element, where
D\scrL ,N =
\biggl\{
s \in \BbbC : \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1
2
, 1 - 1
d\scrL
\biggr)
< \sigma < 1, | t| < N
\biggr\}
,
is considered, however, the proof remains valid for the whole strip D\scrL . Thus, S\scrL is a minimal closed
subset of H(D\scrL ) such that
\^mH
\Bigl(
\^\omega \in \^\Omega : \scrL (s, \^\omega ) \in S\scrL
\Bigr)
= 1. (3.3)
Also, under hypotheses of the theorem, it was proved in [9], Theorem 3.1, that the support of
the Hu(D)-valued random element
\bigl(
\zeta (s, \alpha 1, \omega 1; a11), . . . , \zeta (s, \alpha r, \omega r; arlr)
\bigr)
is the set Hu(D), i.e.,
Hu(D) is a minimal closed subset of Hu(D) such that
\widetilde mH
\Bigl(
(\omega 1, . . . , \omega r) \in \Omega 1 \times . . .\times \Omega r :
\bigl(
\zeta (s, \alpha 1, \omega 1; a11), . . . , \zeta (s, \alpha r, \omega r; arlr)
\bigr)
\in Hu(D)
\Bigr)
= 1.
Combining this with (3.3) and (3.2) gives the assertion of the theorem.
4. Proof of Theorem 1.3. Theorem 1.3 is a consequence of Theorems 2.1 and 3.1, and Merge-
lyan’s theorem on the approximation of analytic functions by polynomials [13], see also [23].
Proof of Theorem 1.3. By the Mergelyan theorem, there exist polynomials p(s) and pjl(s) such
that
\mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| f(s) - p(s)
\bigm| \bigm| < \epsilon
4
(4.1)
and
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq j\leq r
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq l\leq lj
\mathrm{s}\mathrm{u}\mathrm{p}
s\in Kjl
\bigm| \bigm| fjl(s) - pjl(s)
\bigm| \bigm| < \epsilon
2
. (4.2)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
670 R. MACAITIENĖ
Since f(s) \not = 0 on K, we have that p(s) \not = 0 on K as well if \epsilon is small enough. Therefore, there
exists a continuous branch of \mathrm{l}\mathrm{o}\mathrm{g} p(s) on K which is analytic in the interior of K. This and the
Mergelyan theorem show that there is a polynomial q(s) such that
\mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| p(s) - eq(s)
\bigm| \bigm| < \epsilon
4
.
Therefore, (4.1) implies that
\mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| f(s) - eq(s)
\bigm| \bigm| < \epsilon
2
. (4.3)
Define the set
G =
\Biggl\{
(g, g11, . . . , grlr) \in Hv : \mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| g(s) - eq(s)
\bigm| \bigm| < \epsilon
2
, \mathrm{s}\mathrm{u}\mathrm{p}
1\leq j\leq r
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq l\leq lj
\mathrm{s}\mathrm{u}\mathrm{p}
s\in Kjl
\bigm| \bigm| gjl(s) - pjl(s)
\bigm| \bigm| < \epsilon
2
\Biggr\}
.
Then, in view of Theorem 3.1, G is an open neighbourhood of the element
\bigl(
eq(s), p11(s), . . . , prlr(s)
\bigr)
of the support of the measure PZ . Consequently, PZ(G) > 0. Therefore, by Theorem 2.1,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
T\rightarrow \infty
PT (G) \geq PZ(G) > 0,
and, by the definition of G,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
T\rightarrow \infty
1
T
\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}
\Biggl\{
\tau \in [0, T ] : \mathrm{s}\mathrm{u}\mathrm{p}
s\in K
\bigm| \bigm| \scrL (s+ i\tau ) - eq(s)
\bigm| \bigm| < \epsilon
2
,
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq j\leq r
\mathrm{s}\mathrm{u}\mathrm{p}
1\leq l\leq lj
\mathrm{s}\mathrm{u}\mathrm{p}
s\in Kjl
\bigm| \bigm| \zeta (s+ i\tau , \alpha j ; ajl) - pjl(s)
\bigm| \bigm| < \epsilon
2
\Biggr\}
> 0.
Combining this with inequalities (4.3) and (4.2) gives the assertion of Theorem 1.3.
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Received 15.09.14,
after revision — 21.07.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
|
| id | umjimathkievua-article-1585 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T02:08:36Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/85/3e8f8f7c58f0ca83bb7e75f5d3411285.pdf |
| spelling | umjimathkievua-article-15852019-12-05T09:19:33Z Joint universality for $L$-functions from Selberg class and periodic Hurwitz zeta-functions Спiльна унiверсальнiсть для $L$ -функцiй iз класу сельберга та перiодичнi дзета-функцiї Хурвiца Macaitienė, R. Макатіене, Р. We obtain a joint universality theorem on the approximation of a collection of analytic functions by a collection of shifts consisting of $L$-functions from the Selberg class and periodic Hurwitz zeta-functions. Встановлено теорему про спiльну унiверсальнiсть наближення сiм’ї аналiтичних функцiй сiм’єю зсувiв, що складаються з $L$-функцiй iз класу Сельберга та перiодичних дзета-функцiй Хурвiца. Institute of Mathematics, NAS of Ukraine 2018-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1585 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 5 (2018); 655-671 Український математичний журнал; Том 70 № 5 (2018); 655-671 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/1585/567 Copyright (c) 2018 Macaitienė R. |
| spellingShingle | Macaitienė, R. Макатіене, Р. Joint universality for $L$-functions from Selberg class and periodic Hurwitz zeta-functions |
| title | Joint universality for $L$-functions from Selberg class and periodic Hurwitz
zeta-functions |
| title_alt | Спiльна унiверсальнiсть для $L$ -функцiй iз класу сельберга
та перiодичнi дзета-функцiї Хурвiца
|
| title_full | Joint universality for $L$-functions from Selberg class and periodic Hurwitz
zeta-functions |
| title_fullStr | Joint universality for $L$-functions from Selberg class and periodic Hurwitz
zeta-functions |
| title_full_unstemmed | Joint universality for $L$-functions from Selberg class and periodic Hurwitz
zeta-functions |
| title_short | Joint universality for $L$-functions from Selberg class and periodic Hurwitz
zeta-functions |
| title_sort | joint universality for $l$-functions from selberg class and periodic hurwitz
zeta-functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1585 |
| work_keys_str_mv | AT macaitiener jointuniversalityforlfunctionsfromselbergclassandperiodichurwitzzetafunctions AT makatíener jointuniversalityforlfunctionsfromselbergclassandperiodichurwitzzetafunctions AT macaitiener spilʹnauniversalʹnistʹdlâlfunkcijizklasuselʹbergataperiodičnidzetafunkciíhurvica AT makatíener spilʹnauniversalʹnistʹdlâlfunkcijizklasuselʹbergataperiodičnidzetafunkciíhurvica |