1. Introduction
We denote by , , , , , and the sets of positive integers, non-negative integers, prime numbers, rational numbers, real numbers, and complex numbers, respectively, and by a complex variable.
As is well known, the Riemann zeta function
is defined by the Dirichlet series and has a representation by the Euler product over primes, i.e.,
The Hurwitz zeta function
with the real parameter
,
, is given by the series
Both of the functions and have an analytic continuation to the whole complex plane, except for a simple pole at with residue 1. Note that the function has an Euler product over primes only for the cases and , when and .
In the first decade of the twenty-first century, the new type of universality was discovered by H. Mishou [
1] and independently by J. Steuding and J. Sander [
2]. They opened the so-called mixed joint universality in Voronin’s sense for the Riemann and the Hurwitz zeta functions. More precisely, they proved that a pair of analytic functions is simultaneously approximated by shifts of a pair
with transcendental
. Note that the investigations into a universality property for zeta- and
L-functions began in 1975, when the famous paper [
3] on the universality of the Riemann zeta function by S.M. Voronin was published. For the concepts and methods used in the proof of universality and recent results, we refer to the monographs [
4,
5] and survey papers [
6,
7].
For brevity, throughout this paper, we use the following notations and definitions. Let for every real number . We denote by the Lebesgue measure of a measurable set , by the set of all Borel subsets of a topological space S, and by the set of all holomorphic functions on D. For any compact set , we denote by the set of all complex-valued continuous functions defined on K and holomorphic in the interior of K, while by , we denote the subset of , consisting of all elements which are non-vanishing on K.
Theorem 1 (see [
1], Theorem 2)
. Suppose that α is a transcendental number. Let and be compact subsets of with connected complements. Then, for any , , and every , it holds that In 2013, a statement of universality theorems in terms of density was proposed by J.-L. Mauclaire [
8] and independently by A. Laurinčikas and L. Meška [
9]. Such a statement of Theorem 1 was given in [
10]. To be precise, let
Then, the following modification of Theorem 1 was obtained.
Theorem 2 (see [
10], Theorem 2)
. Suppose that the elements of the set are linearly independent over the field of rational numbers . Let , , , and be as in Theorem 1. Then, the limitexists for all but at most countably many . More general results of the same type can be found, for example, in [
11].
The aim of this paper is to show an analogous result to Theorem 2 for rather general classes of zeta functions.
2. Statement of New Result
A generalization of the Hurwitz zeta function
was introduced by A. Laurinčikas [
12]. For the periodic sequence
, with a minimal period
, and a fixed real
,
, the periodic Hurwitz zeta function
is defined by the Dirichlet series
Since the sequence
is a periodic sequence,
From this, we deduce that the function
has an analytic continuation to the whole complex plane except for a simple pole at the point
with residue
.
A class of the Matsumoto zeta functions is a second class under our consideration, particularly, since it covers a wide class of classical zeta functions having the Euler product representation over primes. It was introduced by K. Matsumoto in [
13]. For every
, let
, and for
, with
,
. We denote by
the
mth prime number, and
. We assume that
with a positive constant
and nonnegative constants
and
. We define polynomials
of degree
. The function
is called the Matsumoto zeta function. The product on right-hand side of the equality (
1) converges absolutely for
. In this region, the function
has a Dirichlet series expansion as
with the coefficients satisfying an estimate
for every
if all prime factors of
k are large (for details, see [
14]). For brevity, we define its shifted version by
where
. Then, it is easy to see that
is absolutely convergent for
.
Moreover, we assume that the function satisfies the following conditions:
- (i)
can be meromorphically continued to the region , , and all poles in this region belong to a compact set that has no intersections with the line ;
- (ii)
for and a positive constant ;
- (iii)
It holds the mean-value estimate
All of the functions satisfying the abovementioned conditions construct the class of Matsumoto zeta functions, and we denote the set of all such as .
In 2015, R. Kačinskaitė and K. Matsumoto proved [
15] a mixed joint universality theorem for a wide class of Matsumoto zeta functions and for the periodic Hurwitz zeta function with a transcendental parameter.
For the proof of mixed joint universality, the Bagchi method [
16] can be used, but in the case of the whole class
, it is difficult to prove the denseness lemma. Therefore, we use one more restriction class, namely, the Steuding class
(see [
5]).
We say that belongs to the class if the following assumptions are fulfilled:
- (a)
has a Dirichlet series expansion with for every ;
- (b)
There exists such that can be meromorphically continued to the region and is holomorphic there, except for a pole at ;
- (c)
For any fixed and any , there exists a constant such that ;
- (d)
There exists the Euler product expansion
- (e)
there exists a constant
such that
where
denotes the number of primes
p up to
x.
Let
, and suppose that
is an infimum of all
for which
holds for any
. Then,
, and we see that
.
In 2015, the first result on the mixed joint universality for the tuple
was proved by R. Kačinskaitė and K. Matsumoto (see [
15]). Later, it was proved in a more general situation extending the collection of periodic Hurwitz zeta functions (see [
14]).
Theorem 3 (see [
15], Theorem 2.2)
. Suppose that , and α is a transcendental number. Let be a compact subset of , be a compact subset of , and both have connected complements. Then, for any , , and , it holds that The aim of the present paper is to prove the modification of Theorem 3 in terms of density and to give further certain generalizations.
Now, we state the main result of the present paper.
Theorem 4. Suppose that , and α is a transcendental number. Let , , , and be as in Theorem 3. Then, for all but at most countably many , it holds that Remark 1. The transcendence of α can be replaced by the assumption that the elements of the set are linearly independent over as it shown in Theorem 2.
3. Two Probabilistic Results
For the proof of Theorem 4, the probabilistic approach is used. In this section, we present joint mixed limit theorems on weakly convergent probability measures in the space of analytic functions and a proposition for the support of the probability measure.
Since we are interested in the proof of a joint limit theorem for the tuple
, we deal with more specified regions than
and
(for the arguments, refer to [
15] or [
17]). As is known, the function
has finitely many poles by condition (i) (we denote them by
); then, we put
Since the function
can be written as a linear combination of the Hurwitz zeta functions
, it is entire or has at most a simple pole at
. Let
and
and
be two open regions of
and
, respectively. By
, we mean the Cartesian product of the spaces
and
. Let
, and for
, we define
with
,
,
, and
For the definition of the limit measure, we need a certain probability space. Let
. We define two tori
where
for all
and
for all
, respectively. With the product topology and pointwise multiplication, both tori
and
become compact topological Abelian groups. Therefore, on
and
, there exist the probability Haar measures
and
, respectively. Thus, we obtain the probability spaces
and
. We denote by
the projection of
to the coordinate space
,
; while, for
, let
according to the factorization of
m into the prime divisors
and
,
, the projection to the coordinate space
.
Now, let , and we denote the elements of by . Since is a compact topological Abelian group, we can define the probability Haar measure on . This leads to a probability space .
On
, we define the
-valued random element
by the formula
Here,
,
are
-valued and
-valued random elements defined on
and
, respectively. We denote by
the distribution of the random element
, i.e.,
Now, we are in position to state a mixed joint limit theorem for the tuple of the class of zeta functions.
Theorem 5. Suppose that , and α is a transcendental number. Then, the measure converges weakly to as .
Proof. The proof of this theorem is given in [
15] (Section 3, Theorem 2.1). We only note that the transcendence of
plays an essential role in the proof. □
The second probabilistic result used in the proof of Theorem 4 is that we need to construct an explicit form for the support of the measure
. To obtain the mentioned result, we use the positive density method. Therefore, it is necessary to assume that the function
belongs to the Steuding class
; in particular, the condition (e) must be satisfied (for the details, see [
15] (Section 4, Remark 4.4)).
Let
,
,
,
, and
be as in Theorem 3. Then, there exists a real number
,
and a sufficiently large positive number
M such that
belongs to
Since
, it has only one pole at
; then, we put
. Therefore,
. Analogously, we can find a sufficiently large positive number
N such that
belongs to
Now, if in Theorem 5, we take and , we obtain an explicit form of the ’s support.
Theorem 6. The support of the measure is the set , where .
Proof. The proof of the theorem can be found in [
15] (Lemma 4.3). □
4. Proof of Theorem 4
First, we recall two propositions used in the proof of the main result of the paper.
We recall that a set is said to be a continuity set of the probability measure P if , where is the boundary of A. Note that the set is closed; therefore, it belongs to the class . We are interested in the property of probability measures defined in terms of continuity sets, which is equivalent to weak convergence. Therefore, we use the following fact.
Theorem 7. Let and P be probability measures on . Then, the following assertions are equivalent:
- (1)
converges weakly to P as ,
- (2)
for all continuity sets A of P.
Proof. For the proof, see [
18], (Theorem 2.1). □
We also recall the Mergelyan theorem on the approximation of analytic functions by polynomials.
Theorem 8. Let be a compact subset with connected complement, and let be a continuous function on K analytic inside K. Then, for any , there exists a polynomial such that Proof. The proof of the theorem can be found in [
19]. □
Proof of Theorem 4. Since
on
, by the Mergelyan theorem, there exist polynomials
and
such that, for every
,
In view of Theorem 6, an element belongs to the set S, i.e., to the support of the measure .
Consider the set
This set is an open subset in
and, by Theorem 6, an open neighborhood of an element
. Therefore, by Theorems 5 and 7, the inequality
holds.
Now, for
and
fulfilling the conditions of Theorem 4, we define the set
by
with the boundary
It is easy to see that with different
and
the boundaries
and
are disjoint. Therefore, only countable many sets
can have the positive measure
. Hence,
for at most countable set of values
, i.e.,
is a continuity set of
for all but at most countable many
. Moreover, in view of (
2),
. Therefore, by Theorem 5, we have
for all but at most countable many
. This, together with the definitions of
and
, prove the theorem. □
5. Concluding Remarks
Theorem 4 can be generalized in the following direction.
Suppose that
is a real number such that
, and
is a positive integer,
. Let
. For each
j and
l,
,
, let
be a periodic sequence of complex numbers
with minimal period
, and let
be the corresponding periodic Hurwitz zeta function. We denote by
the least common multiple of periods
. Let
be a matrix consisting of elements
from the periodic sequences
,
,
, i.e.,
Theorem 9. Suppose that are algebraically independent over , , , and belongs to the class . Let be a compact subset of and be a compact subset of , all of them with connected complements. Suppose that and . Then, for all but at most countably many , it holds that Proof. In [
14], the joint mixed universality was proved under the same conditions as in the theorem, instead of “lim”, studying “lim inf” for every
. Therefore, arguing in similar way as in the proof for Theorem 4, we can show the universality inequality (
3).
However, we offer some highlights. Let
, and let
be a polynomial satisfying the second inequality of (
2) for each
,
.
Instead of the set
G in the proof of Theorem 4, we consider the set
and show that
, where
is a distribution of the
-random element constructed for the collections of zeta functions in the theorem. For further details, refer to [
14].
Next we define the set
and obtain that it is a continuity set of the measure
for all but at most countably many
. Again, arguing as for
G and
, we obtain
. Therefore, for all but at most countably many
,
. In view of the similarity of
’s construction to
extending a collection of the periodic Hurwitz zeta functions (for the exact definition of
, see p. 195 in [
14]), this and the definition of
complete the proof. □
Finally, we mention that Theorem 9 can be shown under different conditions than the algebraic independence over
of the parameters
. In particular, we can prove that the universality inequality (
3) holds if the elements of the set
are linearly independent over
.