1. Introduction
Let
be a complex variable, and
be the set of all prime numbers. The Riemann zeta-function is defined, for
, by
The function
with real
s was already known to L. Euler. B. Riemann began to study
with complex
s and applied it to the investigation of the distribution of prime numbers in the set
. The function
has analytic continuation to the whole complex plane, except for the point
, which is a simple pole with residue 1. Riemann’s ideas were successfully realized by J. Hadamard and C.J. de la Valée Poussin at the end of the 19th century. Riemann proved for
the functional equation
where
denotes the Euler gamma function. Moreover, Riemann stated some hypotheses on the zero-distribution of the function
. From Equation (
1), it follows that
,
, and the points
are called trivial zeros of
. Moreover, it is known that the function
has infinitely many of the so-called non-trivial zeros that are complex and lie in the strip
. The famous Riemann hypothesis asserts that all non-trivial zeros of
are located on the critical line
. It is known that more than 41 percent of non-trivial zeros in the sense of density lie on the critical line [
1]. Recently, in [
2], this was improved to more than
percent. There are also other important hypotheses on the value distribution of the function
. For example, by the Lindelöf hypothesis, for every
,
Recall that the notation
,
, means that there exists a constant
such that
. On the other hand, the theory of the function
is sufficiently rich in the final results. One of them is the universality property discovered by S.M. Voronin [
3], which means that a wide class of analytic functions defined in the strip
can be approximated by shifts
,
. More precisely, we denote by
the class of compact subsets of the strip
D with connected complements, and by
with
the class of continuous non-vanishing functions on
K that are analytic in the interior of
K. Then the improved Voronin universality theorem says [
4] that for every
,
and
,
The latter inequality shows that there exists a constant
such that, for sufficiently large
T, the Lebesgue measure of the set
is greater than
. Thus, there are infinitely many shifts in
approximating a given function from the class
. Obviously, the above theorem is useful in the approximation theory of analytic functions, but also has applications in the theory of the function
(functional independence, zero distribution, moment problem); see, for example, [
5,
6] and [
7], respectively, and an informative survey paper [
8].
The above universality theorem has a discrete version [
9]. Denote by
the cardinality of set
A. Then, for the same
K and
as in (
2), and every
and
,
Here
N runs over the set of non-negative integers.
Universality theorems for the function also have their joint versions. In this case, a collection of functions from the class is simultaneously approximated by a collection of different shifts in , for example, by , where satisfy a certain independence condition. In place of traditional shifts and , generalized shifts and are possible with certain functions and .
The function
, as the main ingredient of the functional Equation (
1), plays an important role in the theory of
. This was observed once more by J.-P. Gram in [
10]. Denote by
,
, the increment of the argument of the function
along the segment connecting the points
and
. The function
is increasing and unbounded from above for
, therefore, the equation
has the unique solution
for
. Gram considered the points
in connection with zeros
of
. He observed that each interval
,
, contains
such that
is a zero of
, and conjectured that this is impossible for
. The Gram conjecture was later confirmed by other authors. Moreover, the Riemann-von Mangoldt formula
where
is the number of zeros of
counted the according multiplicities in the region
, implies that
as
, where
are imaginary parts of non-trivial zeros of
. Thus, the sequence
of the Gram points is quite intriguing. A wide survey of the results on the Gram points is given in [
11]. Equation (
3) also offers a unique solution with arbitrary
in place of
n, and this solution is called the Gram function.
In [
12], a joint universality theorem for the Riemann zeta function with shifts involving the powers of the Gram function has been obtained.
Theorem 1 ([
12])
. Suppose that are fixed different positive numbers. For , let and . Then, for every ,Moreover, “lim inf” can be replaced by “lim” for all but at most countably many . The aim of this paper is to obtain a discrete version of Theorem 1, i.e., the joint approximation of analytic functions by using shifts involving the Gram points. It turns out that the discrete case is more complicated, and we have to add the restriction , .
Theorem 2. Suppose that are different fixed positive numbers not exceeding 1. For , let and . Then, for every ,Moreover, “lim inf” can be replaced by “lim” for all but at most countably many . Theorem 2 is weaker than Theorem 1 with respect to numbers . However, discrete universality theorems are sometimes more convenient for applications because of the easier detection of approximating shifts. This is our motivation to consider a discrete version of Theorem 1.
3. Limit Theorems
Denote by
the space of analytic functions on
D endowed with the topology of uniform convergence on compacta, and put
Let
be the Borel
-field of a topological space
. Define the set
where
for all
. The infinite-dimensional torus
, with the product topology and pointwise multiplication, by the classical Tikhonov theorem, is a compact topological Abelian group. Therefore, on
, the probability Haar measure exists. Let
where
for all
. Then, again,
is a compact topological Abelian group, and, on
, the probability Haar measure
can be defined. This gives the probability space
. Note that
is the product of the Haar measures
on
,
. Denote by
the
pth component,
, of an element of
,
, and by
the elements of
. On the probability space
, define the
-valued random element
where
Note that the latter products, for almost all
, are uniformly convergent on a compact subset of the strip
D, see, for example, Theorem 5.1.7 of [
4], or Lemma 4 of [
15]. Denote by
the distribution of the random element
, i.e.,
For brevity, we set
,
,
and, for
, define
This section is devoted to weak convergence for
as
.
Theorem 3. Suppose that are different fixed positive numbers not exceeding 1. Then converges weakly to as .
We divide the proof of Theorem 3 into lemmas. The first of them deals with probability measures on
. For
, define
For the proof of weak convergence for
, we will apply a notion of uniform distribution modulo 1. Recall that a sequence
is called uniformly distributed modulo 1 if, for every subinterval
,
where
is the indicator function of
, and
denotes the fractional part of
.
We will use the Weil criterion on the uniform distribution modulo 1; see, for example, [
16].
Lemma 7. A sequence is uniformly distributed modulo 1 if and only if, for every , The next lemma gives sufficient conditions for uniform distribution modulo 1; see, for example, [
16], Theorem 3.5.
Lemma 8. Let be a function defined for that is l-times differentiable for . If tends monotonically to zero as and if , then the sequence is uniformly distributed modulo 1.
Lemma 9. Suppose that are different fixed positive numbers not exceeding 1. Then converges weakly to the Haar measure as .
Proof. We apply the Fourier transform method. Denote by
,
,
, the Fourier transform of
, i.e.,
where the sign “
” means that only a finite number of integers
are distinct from zero. The definition of
gives
Obviously,
where
. Thus, it remains to consider the case
. In this case, there exists
such that
because the set of logarithms of all prime numbers is linearly independent over the field of rational numbers. Without a loss of generality, we suppose that
, and
. Then
Thus, by Lemma 3,
as
. This shows that
tends monotonically to zero as
, and
Therefore, Lemma 8 implies that the sequence
is uniformly distributed modulo 1. Hence, by Lemma 7 and (
12)
as
. This and (
13) show that
Since the right-hand side of the latter equality is the Fourier transform of the measure
, the lemma is proven. □
Let
The next step of the proof of Theorem 3 is a limit lemma for
Before that, we recall one assertion on the preservation of weak convergence under certain mappings. Let
and
be two spaces, and
a
-measurable mapping, i.e., for every
,
Then every probability measure
P on
defines the unique probability measure
by
It is well known that every continuous mapping
h is
-measurable, and the following useful statement is valid; see, for example, [
17], Theorem 5.1.
Lemma 10. Suppose that , , and P are probability measures on , a continuous mapping, and converges weakly to P as . Then converges weakly to as .
Let, for
,
and
where
Since
, the latter series are absolutely convergent for
with arbitrary finite
.
Consider the mapping
given by
Let
. Then the following statement is valid.
Lemma 11. Suppose that are different fixed positive numbers. Then converges weakly to as .
Proof. By the definition of
, we have
Therefore, for
,
Since the series for
,
, are absolutely convergent, the mapping
is continuous. Therefore, (
14) and Lemmas 9 and 10 prove the lemma. □
The measure
appears in all joint limit theorems for
and other Dirichlet series. The following lemma is known; see, for example, the proof of Theorem 5.4 in [
12].
Lemma 12. converges weakly to as .
Recall one lemma on convergence in distribution (
) of random elements; see, for example, Theorem 4.2 of [
17].
Lemma 13. Suppose that the space is separable, and the -valued random elements and , , are defined on the same probability space with measure μ. Moreover,and, for every ,Then Proof of Theorem 3. Let
be a random variable defined on a certain probability space with measure
and having a distribution
Define two
-valued random elements
and
and denote by
the
-valued random element with distribution
. Then the assertion of Lemma 12 can be written in the form
and, in view of Lemma 11,
Next we need a metric in the space
. Suppose that
is a sequence of embedded compact subsets such that
and every compact set
lies in some
. Such a sequence exists, for example, we can take a sequence of closed rectangles. Then setting
gives a metric in
inducing the topology of uniform convergence on compacta, and
defines a metric in
inducing the product topology.
Now, Lemma 6, together with definitions of the metrics
and
, yields the equality
Therefore, the definitions of random elements
and
show that, for every
,
This equality and relations (
15) and (
16) allow applying Lemma 13 for the random elements
,
and
. Thus, we obtain the relation
and the theorem is proven. □