1. Introduction
Let
be a sequence of complex numbers, and
be a complex variable. In analytic number theory, Dirichlet series
are very important analytic objects. The latter series are of ordinary type, general Dirichlet series
where
is an increasing to
sequence of real numbers, are also studied. The majority of the so-called zeta- and
L-functions, including the Riemann zeta-function
and Dirichlet
L-functions
where
is a Dirichlet character, whose analytic properties play the crucial role for investigation of prime numbers in the set
and arithmetic progressions, respectively, are defined by Dirichlet series.
Without a class of Dirichlet
L-functions, there are several classes of Dirichlet series cultivated in analytic number theory. Among them, the classes of Hurwitz-type zeta-functions, Lerch zeta-functions, Matsumoto zeta-functions, Epstein zeta-functions, and others. The famous number theorist A. Selberg in [
1] introduced the class
of Dirichlet series including some classical number theoretical zeta- and
L-functions, and stated hypotheses on that class. The Selberg class became an object of numerous studies. We recall the hypotheses which satisfy the functions
of the class
. As usual, we denote by
the Euler gamma-function.
- 1.
For every , the estimate is valid.
- 2.
There exists an integer such that is an entire function of finite order.
- 3.
The function
satisfies the functional equation
where
with positive numbers
q and
, and complex numbers
w and
such that
and
, and
denotes the complex conjugate of
s.
- 4.
The function
has the representation
over the prime numbers with
and
,
.
Much attention is devoted to the structure of the class
. For results, see Section 6.1 of [
2]. In the theory of the class
, the degree of the function
defined as
occupies an important place. For example, it is known that if
, then
, while if
, then
are the Riemann zeta-function, or shifted Dirichlet
L-function
,
. There exists a conjecture that the class
consists of all automorphic
L-functions. For example,
L-functions of normalized holomorphic new forms have a degree
.
In this paper, we are interested in the universality of functions of the class
, i.e., on the approximation of a whole class of analytic functions by shifts
,
,
. Recall that the universality property for
which is a member of
was obtained by S.M. Voronin in [
3]. For an improved version of the Voronin theorem, we use the following notation. Let
, and let
denote the Lebesgue measure of a measurable set
. Suppose that
is a compact set with connected complement, and
is a continuous non-vanishing function on
K and analytic in the interior of
K. Then, for every
,
(see [
2,
4,
5,
6]). A similar assertion also is true for all Dirichlet
L-functions.
The first universality result related to the class
was obtained by J. Steuding in [
2]. Let, for
,
and
. In addition to the hypothesis 4 of the class
, it was required the existence of a polynomial Euler product
Moreover, one more arithmetic condition
with a certain positive
and
was used. Denote by
the class of compact subset of the strip
with connected complements, and by
,
, the class of continuous non-vanishing functions on
K that are analytic in the interior of
K. Denote by
the class satisfying hypotheses of the class
, and (
1) and (
2). Then, the following theorem is true [
2].
Theorem 1. Suppose that . Let and . Then, for every , We note that the class
consists of all functions satisfying axioms 2 and 3 of class
, and (
1) and (
2).
In [
7], Theorem 1 was improved, namely, the condition (
1) was not used. More precisely, Theorem 1 is valid for
satisfying (
2).
For zeta- and
L-functions, the joint universality also is considered. In this case, a collection of analytic functions is simultaneously approximated by a collection of shifts of zeta- or
L-functions. The first result in this direction also belongs to S.M. Voronin. In [
8], he obtained the joint universality for Dirichlet
L-functions with nonequivalent characters and applied it for a theorem on joint functional independence of
L-functions. More general results on joint universality were obtained for the periodic and periodic Hurwitz zeta-functions as well as for Matsumoto zeta-function (see, for example, [
9,
10,
11,
12]). Joint universality theorems also can be proved using only one zeta- or
L-function with different shifts. Our aim is to obtain a joint universality theorem for functions from the Selberg–Steuding class
of functions belonging to the class
and satisfying the condition (
2). The main result of the paper is the following theorem.
Theorem 2. Suppose that , and real algebraic numbers are linearly independent over the field of rational numbers . For , let and . Then, for every , Moreover, “liminf” can be replaced by “lim” for all but at most countably many .
The proof of Theorem 2 is based on weak convergence of probability measures in the space of analytic functions.
2. Limit Lemmas on a Group
We start to consider the weak convergence of probability measures with a case of one compact group. Denote by
the Borel
-field of a topological space
, and define the set
where
denotes the set of all prime numbers, and
for all
. By the classical Tikhonov theorem, the infinite-dimensional torus
, with the product topology and operation of pairwise multiplication, is a compact topological Abelian group. Define one more set
where
for all
. Then, again, by the Tikhonov theorem,
is a compact topological Abelian group. Therefore, on
, the probability Haar measure
can be defined. This gives the probability space
. For
, denote by
the
pth component of an element
,
, and by
the elements of
. Let, for brevity,
.
Now, we will prove a limit lemma on weak convergence for
, as
. For its proof, we apply the following result of A. Baker (see [
13]).
Lemma 1. Suppose that the logarithm of algebraic numbers are linearly independent over . Then, for any algebraic numbers not all simultaneously zero, the inequalitywhere h is the maximum of the heights of the numbers , and C is an effective constant depending on r, and the maximum of the powers of the numbers , is valid. Lemma 2. Suppose that are real algebraic numbers linearly independent over . Then, converges weakly to the Haar measure as .
Proof. For the proofs of weak convergence of probability measures on groups, it is convenient to use the method of Fourier transforms. Thus, denote by
,
,
, the Fourier transform of
, i.e.,
where the star * shows that only a finite number of integers
is distinct from zero. By the definition of
, we have
Obviously,
where
is a collection consisting from zeros. Now, suppose that
. Let, for brevity,
,
where
In this case, there exists
j such that
. Therefore,
are not all zero. Since the numbers
are linearly independent over
, the algebraic numbers
are not all simultaneously zero. It is well known that the set
is linearly independent over
. Therefore, for
, Lemma 1 is applicable, and we obtain that
. Hence, integrating in (
3), we find
This together with (
4) shows that
and the lemma is proved because the right-hand side of the last equality is the Fourier transform of the Haar measure
. □
We will apply Lemma 2 to obtain a joint limit lemma in the space of analytic functions for absolutely convergent Dirichlet series. Denote by
the space of analytic on
functions equipped with topology of uniform convergence on compacta, and set
Let
be a fixed number,
and
Since
and
is decreasing exponentially with respect to
m, the latter series is absolutely convergent in any half-plane
. Extend the functions
,
,
, to the set
of all positive integers by
where
means that
but
, and define
the series also being absolutely convergent for
. Define
and
by
. Since the series
,
, are absolutely convergent in any half-plane, the mapping
is continuous. Therefore, every probability measure
P on
defines the unique probability measure
on
, where
For
, define
where
Moreover, a property of preservation of weak convergence under continuous mappings (see, for example, Theorem 5.1 of [
14]), leads to the following lemma.
Lemma 3. Suppose that are real algebraic numbers linearly independent over . Then, converges weakly to the measure as .
Proof. By the definitions of
and
, and the mapping
, for every
, we have
Thus,
. Thus, the continuity of
, Lemma 2 and Theorem 5.1 of [
14] prove the lemma. □
Consider one more measure
Lemma 4. Suppose that are real algebraic numbers linearly independent over . Then, with every also converges weakly to the measure as .
Proof. Define the mapping
by
. Then, the mapping
remains continuous, and repeating the arguments of the proof of Lemma 3, we obtain that
converges weakly to the measure
as
. By the definitions of
and
, we have
with
. At this moment, we use the invariance of the Haar measure
, i.e., that
for all
and
. Thus, we find
□
3. Limit Theorems
In this section, we will prove a joint limit theorem for the function
from class
. More precisely, we will consider the weak convergence for
where
as
. For the proof, we will apply Lemmas 3 and 4, some ergodicity results, and estimates for difference
. We start with the latter problem.
Recall the metric in the space
. For
, define
Here,
is a sequence of compact embedded sets such that
and each compact set
lies in
for some
l. Then,
is a metric in
inducing the topology of uniform convergence on compacta. For
,
, taking
we have a metric in
inducing the product topology.
Lemma 5. Suppose that are arbitrary real numbers. Then, Proof. Let the number
come from the definition of
, and
Then, the Mellin formula
implies the representation (see, for example, [
2])
where
. Hence, by the residue theorem,
where
and
Let
be an arbitrary compact set. We fix
such that
for all
, and put
. Then,
for
. This and equality (
2), for
and
, gives
Taking
v in place of
, we have, for
,
It is known [
2] that, for fixed
,
This, for the same
and
, gives
Using the well-known estimate
we find that, for all
,
Similarly, by (
9), for
,
The latter estimate, (
10) and (
7) prove that, for every compact set
,
Therefore, the lemma follows from the definitions of the metrics and . □
Now, for
, let
where
Then, it is known [
2] that the latter series, for almost all
, are uniformly convergent on the compact subset of the half-plane
. Since the Haar measure
is the product of the Haar measures
on
, we have that
is the
-valued random element. Moreover, an analogue of Lemma 5 is valid.
Lemma 6. Suppose that are arbitrary real numbers. Then, for almost all , Proof. It is known [
2] that, for almost all
,
Therefore, repeating the proof of Lemma 5, we obtain that, for a compact set
and real number
a,
with certain
. In this case, in the analogous of estimate (
7), we have not the second term on the right-hand side. Since
, estimate (
11) and the definitions of the metrics
and
prove the lemma. □
Now, we are ready to consider the measure .
Theorem 3. Suppose that real algebraic numbers are linearly independent over . Then, on , there exists a probability measure P such that converges weakly to P as .
Proof. Recall that a family of probability measures
on
is called tight if, for every
, there exists a compact set
such that
for all
Q.
Denote by
marginal measures of the measure
,
. Since the series for
is absolutely convergent, we obtain by a standard way that the sequence
is tight,
. Then, for every
, there exists a compact set
such that, for all
,
Let
. Then,
K is a compact set in
. Moreover, by (
12), for all
,
Thus,
for all
. Hence, the sequence
is tight. Therefore, by the Prokhorov theorem, see [
14], the sequence
is relatively compact. This means that every sequence of
contains a subsequence
weakly convergent to a certain probability measure
P on
as
.
Denote by
the
-valued random element having the distribution
, and by
the convergence in distribution. Then, we have
On the certain probability space with measure
, define the random variable
which is uniformly distributed on
. Moreover, let
and
By Lemma 3,
and Lemma 5 implies, for every
,
This together with relations (
13) and (
14) shows that all hypotheses of Theorem 4.2 from [
2] are satisfied. Therefore,
and this proves the theorem. □
By (
15), the measure
P is independent on the sequence
. Since the sequence
is relatively compact, it follows that
On
, define one more measure
for almost all
. Then, by (
15), Lemmas 4 and 6, similarly as above, we obtain the analogue of Theorem 3.
Theorem 4. Suppose that real algebraic numbers are linearly independent over . Then, also converges weakly to the measure P as .
4. Identification of the Measure P
For the proof of Theorem 2, the explicit form of the limit measure in Theorems 3 and 4 is needed. For this, some elements of ergodic theory can be applied.
For brevity, we set
and define
Then, is a measurable measure preserving transformation of the group , and form a group of these transformations. For , let . If the sets A and differ one from another at most by a set of -measure zero, then the set A is called invariant. All invariant sets form a -field. If this field consists only of sets of -measure 1 or 0, then the group is called ergodic.
Lemma 7. Suppose that real algebraic numbers are linearly independent over . Then, the group is ergodic.
Proof. The characters
of the group
are of the form
where the sign * means that only a finite number of integers
is not zero. This already was used in the proof of Lemma 2 for the definition of the Fourier transform of the measure
. Suppose that
A is an invariant set with respect to
, and
is a nontrivial character of
, i.e.,
. Then, by (
17),
, and thus
in the notation used in the proof of Lemma 2. Therefore, there exists a real number
such that
Take the indicator function
of the set
A. In virtue of invariance of the set
A, we have
for almost all
. Hence, denoting by
the Fourier transform of a function
g, we find
Therefore, in view of (
18),
Now, suppose that
denotes the trivial character of
, and
. Then, taking into account (
19), we have
for an arbitrary character
of
. This shows that
for almost all
. However,
is the indicator function of the
A, thus,
or
for almost all
. In other words,
or
, and the lemma is proved. □
Denote by
the distribution of the
-valued random element
, i.e.,
Lemma 8. The measure P in Theorems 3 and 4 coincide with .
Proof. Suppose that
A is a continuity set of the measure
P, i.e.,
, where
denotes the boundary of
A. Then, Theorem 4 together with the equivalent of weak convergence of probability measure in terms of continuity sets (see, for example, Theorem 2.1 of [
14]) yields
On
, define the random variable
Lemma 7 implies the ergodicity of the random process
. Therefore, by the Birkhoff–Khintchine ergodic theorem (see, for example, [
15]), we obtain
where
is the expectation of
. However, by the definitions of
and
,
Thus, in view of (
20),
for all continuity sets
A of
P. It is well known that continuity sets constitute the defining class. Thus,
. □
It remains to find a support of the measure . We recall that the support of is a minimal closed set such that .
Lemma 9. The support of the measure is the set .
Proof. It is known that the support of the measure
is the set
(see [
2] or [
7]). Since the space
is separable, we have
Therefore, it suffices to consider the measure
on rectangular sets
Moreover,
. These remarks show that
This and the minimality of the support prove the lemma. □
6. Concluding Remarks
Let be a function from the Selberg–Steuding class. Combining algebraic, analytic, and probabilistic methods, we obtain a theorem on simultaneous approximations of a collection of analytic functions in the strip by a collection of shifts , where are real algebraic numbers linearly independent over the field of rational numbers, and is a certain number depending on L. More precisely, we proved that the set of the above shifts has a positive lower density, or even positive lower density for all but not most countable many accuracies of approximation. Thus, the set of approximating shifts is infinite.
It is important that the measure is independent of the used shift and converges weakly to the measure . This result can be used for the proof of other joint theorems on joint approximation of analytic functions by more complicated shifts including discrete shifts.