Abstract:
In the paper, a joint discrete universality theorem for periodic zeta-functions with multiplicative coefficients on the approximation of analytic functions by shifts involving the sequence 
. of imaginary parts of nontrivial zeros of the Riemann zeta-function is obtained. For its proof, a weak form of the Montgomery pair correlation conjecture is used. The paper is a continuation of [A. Laurincˇikas, M. Tekore˙, Joint universality of periodic zeta-functions with multiplicative coefficients, Nonlinear Anal. Model. Control, 25(5):860–883, 2020] using nonlinear shifts for approximation of analytic functions.
Keywords: joint universality, nontrivial zeros of the Riemann zeta-function, periodic zeta-function, space of analytic functions, weak convergence.
Article
Joint universality of periodic zeta-functions with multiplicative coefficients. II
Laurinˇcikas Antanas antanas.laurincikas@mif.vu.lt
Šiauciunas Darius darius.siauciunas@sa.vu.lt
Tekore Monika monika.tekore@mif.stud.vu.lt
Recepción: 19 Julio 2020
Revisado: 19 Febrero 2021
Publicación: 01 Mayo 2021
It is well known that some zeta- and .-functions, and even some classes of Dirichlet series, for example, the Selberg-Steuding class, see [29, 32], are universal in the Voronin sense, i.e., a wide class of analytic functions can be approximated by one and the same zeta-function. For example, in the case of the Riemann zeta-function
, analytic nonvanishing functions on the strip
are approx- imated by shifts
(continuous case), or shifts
(discrete case); see [1, 6, 13, 24, 32].
The above shifts are very simple, T and kh occur in them linearly. It turned out that the approximation remains valid also with more general shifts. A significant progress in this direction was made by Pan´kowski [31] using the shifts .
with .
and a wide class of reals a and B. The papers [22] and [35] are also devoted to approximation of analytic functions by generalized shifts of zeta-functions. In [5], the shifts .
were applied, where
is the sequence of imaginary parts of nontrivial zeros of the Riemann zeta-function.
Universality in the Voronin sense also has its joint version. In the joint case, a col- lection of analytic functions is approximated simultaneously by a collection of shifts of zeta- or .-functions. The first joint universality theorem belongs to Voronin who proved [36] the joint universality of Dirichlet .-functions
Ob- viously, in joint universality theorems, the approximating shifts must be in some sense independent. Voronin required [36] for this the pairwise nonequivalence of Dirichlet characters, i.e., in fact, he considered joint universality of different Dirichlet .-functions. On the other hand, as it was observed by Pan´kowski [31], the independence of approxi- mating shifts of Dirichlet .-functions can be ensured by different functions
in shifts
even with the same characters χj. This obser- vation extends significantly classes of jointly universal functions. For example, the joint universality with generalized shifts was obtained in [16] and [20].
In general, joint universality of zeta-functions was widely studied, and many results are known; see, for example, general results obtained in [7–11,14,26,30] and other papers by authors of the mentioned works. In this note, we focus on joint universality of so- called periodic zeta-functions with generalized shifts involving the sequence
. N of imaginary parts of nontrivial zeros of the function
. We will mention some joint universality results involving the latter sequence. Note that the behaviour of the sequence
, as of nontrivial zeros of 
is very complicated, and at the moment, its known properties are not sufficient for the proof of universality. Therefore, in [5], the conjecture that, for c > 0,
(1)was introduced. This conjecture is inspired by the Montgomery pair correlation conjecture
[28] that
where
. are arbitrary real numbers, and
Now we will state a joint universality theorem for Dirichlet .-functions involving the sequence
obtained in [18]. Denote by the class of compact subsets of the strip . with connected complements, and by .
the class of continuous nonva- nishing functions on . that are analytic in the interior of K.
Suppose that χ., . . . , χ. are pairwise nonequivalent Dirichlet characters, and estimate (1)is true.
Then, for every ε > 0 and h > 0,
Moreover “lim inf” can be replaced by “lim” for all but at most countably many ε > 0.
Here #A denotes the cardinality of the set ., and . runs over the set N.
Now we recall the definition of the periodic zeta-function, which is an object of in- vestigation of the present note.
be a periodic sequence of complex numbers with minimal period 
Then the periodic zeta-function .
is defined, for σ > 1, by the Dirichlet series
and has an analytic continuation to the whole complex plane, except for a simple pole at the point s = 1 with residue
The sequence . is called multiplicative if
for all coprimes m, n ∈ N. If 0 < α ≤ 1 is a fixed number, then the function
and its meromorphic continuation are called the periodic Hurwitz zeta-function. In [15] and [3], under hypothesis (1), joint universality theorems involving sequence γ. for the pair consisting from the Riemann and Hurwitz zeta-functions and their periodic ana- logues, respectively, were obtained, while in [23], such theorems were proved for Hurwitz zeta-functions.
For
be a periodic sequences of complex numbers with minimal period
be the corresponding zeta-function. The main result of the paper is the following theorem.
Suppose that the sequences
are multiplicative,
are positive algebraic numbers linearly independent over the field of rational numbers, and estimate (1)is true.
Then, for every ε > 0,
Moreover “lim inf” can be replaced by “lim” for all but at most countably many ε > 0.
In [21], joint continuous universality theorems for periodic zeta-functions with shifts defined by means of certain differentiable functions were obtained.
From t
he functional equation for the Riemann zeta-function
it follows that
and the zeros .
are called trivial. Moreover, it is known that 
has infinitely many of so-called complex nontrivial zeros 
lying in the strip
. The famous Riemann hypothesis, one of seven Millennium problems, asserts that β. = 1/2, i.e., all nontrivial zeros lie on the critical line
There exists a conjecture that all nontrivial zeros of
are simple.
We recall some properties of the sequence
By the definition, a sequence {xk: K ∈ N} ⊂ R is called uniformly distributed modulo 1, if, for every subinterval (a, b] ⊂ (0, 1],
where I(a,b] is the indicator function of (a, b], and {u} denotes the fractional part of u ∈ R. Though the sequence {γ.} is distributed irregularly, the following statement is true for it.
The sequence 
. N with every a 
is uniformly distributed modulo 1
Proof. Proof of the lemma is given in [33], and in the above form, was applied in [5].
For convenience, we recall the Weyl criterion on the uniform distribution modulo 1; see, for example, [12].
A sequence
is uniformly distributed modulo . if and only if, for every 
,
Obviously, the uniform distribution modulo 1 of the sequence shows its nonlinear character.
The following statement is well known; see, for example, [34].
For k → ∞,
Denote by H(D) the space of analytic functions on D endowed with the topology of uniform convergence on compacta. We will derive Theorem 2 from a limit theorem on the weak convergence of probability measures in the space
Therefore, we start with a certain probability model.
Let B (X) be the Borel .-field of the space X, and P denote the set of all prime numbers. Define
where
for all 
Then . is a compact topological Abelian group. Moreover, let
where
Then again Ωris a compact topological Abelian group.
Therefore, on (Ωr, B(Ωr)), the probability Haar measure
can be defined. This gives the probability space
Denote by w(p) the .th component, 
of an element 
For brevity, let
and on the probability space
, define the Hr(D)-valued random element
Where
Note that the latter products, for almost all ω., are uniformly convergent on compact subsets of the strip D. Since the periodic sequences
are bounded, the proofs of the above assertions completely coincides with those of Lemma 5.1.6 and The- orem 5.1.7 from [13]. More general results are given in [1]. Denote by 
the distribution of the random element
,
Put .
, define
where
In this section, we will prove the following limit theorem.
Suppose that the sequences ar, . . . , ar are multiplicative, h1, . . . , hr are positive algebraic numbers linearly independent over , and estimate (1)is valid. Then PN converges weakly to
.
We start the proof of Theorem 3, as usual, with a limit lemma in the space Ωr. In this lemma, the uniform distribution modulo 1 of the sequence
and the property of the numbers h1, . . . , hressentially are applied.
For A ∈ B(Ωr), define
Before the statement of a limit theorem for QN , we recall one result of Diophantine type.
Suppose that λ1, . . . , λr . are algebraic numbers such that the logarithms
are linearly independent over
. Then, for any algebraic numbers
, not all zero, we have
where H is the maximum of the heights of β0, β1, . . . , βr, and C is an effectively com- putable number depending on r and the maximum of the degrees of β., β., . . . , β..
The lemma is the well-known Baker theorem on logarithm forms; see, for example [2].
Suppose that h1, . . . , hr are real algebraic numbers linearly independent over
. Then QN converges weakly to the Haar measure
.
Proof. As usual, we apply the Fourier transform method. The characters of the group Ωr
are of the form
where the star “*” shows that only a finite number of integers kjpare distinct from zero. Therefore, the Fourier transform of QNis
where .
Thus, by the definition of QN,
(2)Obviously,
(3)Now, suppose that .
Then there exists j ∈ {1, . . . , r} such that
. Thus, there exists a prime number . such that kjp/= 0. Define
Then, in view of a property of the numbers .
, we have ap= 0. The numbers apare algebraic, and the set 
is linearly independent over
. Therefore, by Lemma 4,
Hence, in virtue of Lemma 1, the sequence
is uniformly distributed modulo 1. This, together with (2) and Lemma 2, shows that, in the case
Thus, in view of (3),
and the lemma is proved because the right-hand side of the latter equality is the Fourier transform of the Haar measure
.
Lemma 5 implies a limit lemma in the space Hr(D) for absolutely convergent Dirich- let series. Let, for a fixed θ > 1.2,
and
Then the latter series are absolutely convergent for σ > 1.2. Actually, since
with every L > 0, the latter series are absolutely convergent even in the whole
complex plane. For B(Hr(D)), define
where
Moreover, let
and let 
be given by the formula
Suppose that h1, . . . , hr are real algebraic numbers linearly independent over
. Then VN,n, as N → ∞, converges weakly to a measure
where
Proof. Since the series for ζn(s, ωj. aj) are absolutely convergent for σ > 1.2, the function u. is continuous, hence (B(Ωr),B(Hr(D)))-measurable. Therefore, the mea- sure Vnis defined correctly. The definitions of QN , VN,nand unimply the equality
Therefore, the lemma follows from Lemma 5 and a preservation of weak convergence under continuous mappings; see [4, Thm. 5.1].
The limit measure Vnin Lemma 6 is independent on h and
and has a good convergence property, which is the next lemma.
Suppose that the sequences a1, . . . , ar are multiplicative. Then Vn converges weakly to Pc as n → ∞.
Proof. In [17], the weak convergence for
was considered, and it was obtained its weak convergence to
and that Vnalso converges weakly to
In other words, Vn and PˆT have the same limit measure Pc.
In view of Lemma 7, to prove Theorem 3, it suffices to show that PN, as N → ∞, and Vn, as n → ∞, have a common limit measure. For this, a certain closeness of
and .
is needed.
There exists a sequence
of compact subsets such that
for all l ∈ N, and if K ⊂ D is a compact set, then K ⊂ K1for some l. Then, putting, for g1, g2 ∈ H(D),
we have a metric in H(D) inducing its topology of uniform convergence on compacta.
Hence,
is a metric in Hr(D) inducing its product topology. Note that, in the proof of the next lemma, the multiplicativity of the sequences
is not used.
Suppose that estimate (1)is true. Then, for every positive h1, . . . , hr and a1, . . . , ar,
(4)Proof. By the definitions of the metrics p and p, it is sufficient to show that, for every compact set K ⊂ D,
(5)j = 1, . . . , r. The equality of type (5) was already used in [3], therefore, only for fullness, we give remarks on its proof.
Thus, let h > 0 and . be arbitrary. We consider .
and
.
Let 0 be as in the definition of vn(m). Then the representation
where
is valid. Hence, for
,
(6)where
and a is the residue of 
at the point s = 1. Let K ⊂ D be an arbitrary compact set, and ε > 0 be such that
Then, in view of (6), for
,
Hence, taking t in place of t + v and
, we have
(7)where
and
Estimate (1) is applied for estimation of the first factor of the integrated function in the integral I. It is well known that, for T ∈ R,
(8)The same estimate is also true for the derivative of
and
Then, in view of (1) and Lemma 3,
This, (6) and an application of the Gallagher lemma connecting discrete and continuous
mean squares for some function, see Lemma 1.4 of [27], give
Therefore, the classical estimate for the gamma-function and the definition of
show that
This, together with (7), proves (5), thus (4).
Proof of Theorem 3. We will use the random element language. Denote by
the Hr(D)-valued random element having the distribution Vn, where Vnis the limit measure in Lemma 6. Then, by Lemma 7,
(9)where 
means the convergence in distribution. Now, let the random variable ηNbe defined on a certain probability space with a measure u, and
Define the Hr(D)-valued random element
Then, in virtue of Lemma 7,
Let
Then Lemma 8 implies that, for every ε > 0,
Therefore, this, (9), (10) and Theorem 4.2 of [4] show that
and the theorem is proved.
We start with the explicit form of the support of the measure Pc. Recall that the support of a probability measure P is a minimal closed set S. such thatP (Sp) = 1.
The support of the measure Pcis the set Sr.
Proof. The space Hr(D) is separable. Therefore [4],
From this it follows that it suffices to consider the measure
on the rectangular sets
Denote by mjHthe Haar measure on Ωj, j = 1, . . . , r. Then the Haar measure
is the product of the measures m1H, . . . , mrH. These remarks imply the equality
(11)It is known [19] that the support of
is the set S. Therefore, (11) and the minimality of the support prove the lemma.
Proof of Theorem 2. The theorem is corollary of Theorem 3, the Mergelyan theorem on the approximation of analytic functions by polynomials [25], and Lemma 9, and it is standard. By the Mergelyan theorem, there exist polynomials
such that
(12)In view of Lemma 9, the set
is an open neighbourhood of an element of the support of the measure
Hence,
(13)Therefore, by Theorem 3 and the equivalent of weak convergence of probability measures in terms of open sets,
This, the definitions of P. and G., together with inequality (12), prove the first part of the theorem.
For the proof of the second part of the theorem, we define one more set
Then 
is a continuity set of the measure
for all but at most countably many ε > 0, moreover, in view of (12), the inclusion
is valid. Therefore, Theorem 3, the equivalent of weak convergence of probability measures in terms of continuity sets and (13) lead the inequality.
for all but at most countably many ε > 0. This, the definitions of PNand 
prove the second part of the theorem.
