On Mixed Joint Discrete Universality for a Class of Zeta-Functions: a Further Generalization

We present the most general at this moment results on the discrete mixed joint value-distribution (Theorems 5 and 6) and the universality property (Theorems 3 and 4) for the class of Matsumoto zeta-functions and periodic Hurwitz zetafunctions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.


Introduction
In the series of previous works by the authors (see [9,10,11,12]), the tuple consisting from the wide class of the Matsumoto zeta-functions ϕ(s) and the periodic Hurwitz zeta-functions ζ(s, α; B) is considered in the frame of the studies on the value distribution of such pair, as well as the universality property and the functional independence.
Let s = σ + it be a complex variable and by P, N, Q, R and C denote the set of all prime numbers, positive integers, rational numbers, real numbers and complex numbers, respectively. Suppose that α, 0 < α ≤ 1, is a fixed real number. Recall the definitions of both the functions under our interest -the Matsumoto zeta-functions ϕ(s) and the periodic Hurwitz zeta-functions ζ(s, α; B).
For m ∈ N, let g(m) ∈ N, f (j, m) ∈ N, 1 ≤ j ≤ g(m). Denote by p m the mth prime number, and let a The function ϕ(s) converges absolutely in the region σ > α + β + 1, and, in this region, it can be written as the Dirichlet series c k k s+α+β with coefficients c k such that c k = O k α+β+ε if all prime factors of k are large for every positive ε (for the comment, see [11]).
Suppose that the function ϕ(s) satisfies the following conditions: (i) ϕ(s) can be continued meromorphically to σ ≥ σ 0 , where 1 2 ≤ σ 0 < 1, and all poles of ϕ(s) in this region are included in a compact set which has no intersection with the line σ = σ 0 ; (ii) for σ ≥ σ 0 , ϕ(σ + it) = O(|t| C2 ) with a certain C 2 > 0; (iii) it holds the mean-value estimate We call the set of all such functions ϕ(s) as the class of Matsumoto zetafunctions and denote by M. Now, for N 0 := N ∪ {0}, let B = {b m : m ∈ N 0 } be a periodic sequence of complex numbers b m with minimal positive period l ∈ N. The periodic Hurwitz zeta-function ζ(s, α; B) with parameter α ∈ R, 0 < α ≤ 1, is given by the Dirichlet series It is known that the function ζ(s, α; B) is analytically continued to the whole complex plane, except for a possible simple pole at the point s = 1 with residue b := 1 l l−1 m=0 b m . If b = 0, then ζ(s, α; B) is an entire function. It is possible to prove functional limit theorems for the whole class M, but it is difficult to prove the denseness lemma which is necessary to prove the universality theorem. Therefore in the proof of the universality we use an assumption that the function ϕ(s) belongs to the Steuding class S defined below.
We say that the function ϕ(s) belongs to the class S if the following conditions are satisfied: (b) there exists σ ϕ < 1 such that ϕ(s) can be meromorphically continued to the half-plane σ > σ ϕ and is holomorphic except for a pole at s = 1; (c) for every fixed σ > σ ϕ and ε > 0, the order estimate ϕ(σ+it) = O(|t| C3+ε ) with C 3 ≥ 0 holds; (d) there exists the Euler product expansion over primes, i.e., (e) there exists a constant κ > 0 such that where π(x) denotes the number of primes p up to x.
For ϕ(s) ∈ S, let σ * be the infimum of all σ 1 such that holds for any σ ≥ σ 1 . Then 1 2 ≤ σ * < 1, and we see that S ⊂ M if σ 0 = σ * + ε is chosen. Note that the class S is not a subclass of the Selberg class! To formulate mixed joint discere limit theorems and universality property for the tuple ϕ(s), ζ(s, α; B) and further results, we need some notation. For any compact set K ⊂ C, denote by H c (K) the set of all C-valued functions defined on K, continuous on K and holomorphic in the interior of K. By H c 0 (K) denote the subset of H c (K) such that, on K, all elements of H c (K) are non-vanishing. Let D(a, b) = {s ∈ C : a < σ < b} for a, b ∈ R, a < b, and we denote by meas{A} the Lebesgue measure of the measurable set A ⊂ R. For any set S, B(S) denotes the set of all Borel subsets of S, and, for any region D, H(D) denotes the set of all holomorphic functions on D.
The first result on the mixed joint universality of the pair ϕ(s), ζ(s, α; B) is the following theorem, which considers the situation when the shifting parameter is moving continuously.
Theorem 1 [Theorem 2.2, [9]]. Suppose ϕ(s) ∈ S, and α is a transcendental number, 0 < α < 1. Let K 1 be a compact subset of the strip D σ * , 1 , K 2 be a compact subset of the strip D 1 2 , 1 , both with connected complements. Suppose . Then, for every ε > 0, it holds that More interesting and complicated questions on mixed joint universality are concerning the discrete case, when shifting parameters take discrete values. The first result of such kind is the discrete analogue of Theorem 1, which was proved by the authors in [10].
Let h > 0 be the shifting parameter, that is the common difference of the relevant arithmetic progression, and put Remark 1. In [12], we wrote the set L(α, h) in a slightly different way: However this notation may cause a misunderstanding, because if we choose h = 2π/ log(m 0 + α) with a certain m 0 ∈ N 0 , then L(α, h) = {log p : p ∈ P} ∪ {log(m + α) : m ∈ N 0 }, and even if the elements of this set are linearly independent, the required condition on (1.1) cannot be satisfied. This matter was pointed out by Professor Antanas Laurinčikas, to whom the authors express their sincere thanks.
Theorem 2 [Theorem 3, [10]]. Suppose ϕ(s) ∈ S, and that the set L(α, h) is linearly independent over Q. Let K 1 , K 2 , f 1 (s) and f 2 (s) be the same as in Theorem 1. Then, for every ε > 0, it holds that Here and in what follows, #{A} denotes the cardinality of the set A.
In [12], we have extended our investigations and studied the case when common differences of arithmetic progressions for both of the zeta-functions in the tuple are different.
The aim of this paper is to show more general results than those mentioned above. Here we prove two mixed joint universality theorems -in the cases of the same and different common differences of arithmetic progressions. The novelty is that a wide collection of the periodic Hurwitz zeta-functions ζ(s, α j ; B jl ) will be constructed; here for each α j a collection of sequences B jl is attached. This type of general collection of periodic Hurwitz zeta-functions has been studied in several previous articles (such as [4,11,16,22] and [7]) in the continuous case. In the discrete case, there are papers for periodic zeta-functions (see [18,19]). The results in the present article are discrete analogous of Theorem 4.2 from [11], but in [11] we assume the stronger hypothesis, that is the algebraic independence of the parameters α 1 , ..., α r .
Our first new discrete mixed joint universality theorem is as follows.
Theorem 3. Suppose that the set L(α, h) is linearly independent over Q, rankB j = l(j), j = 1, ..., r, and ϕ(s) belongs to the class S. Let K 1 be a compact subset of D(σ * , 1), K 2jl be compact subsets of D( 1 2 , 1), l = 1, ..., l(j), all of them with connected complements. Suppose that f 1 ∈ H c 0 (K 1 ) and f 2jl ∈ H c (K 2jl ). Then, for every ε > 0, it holds that In Theorem 3, there appears only one common difference h. Our second new theorem on universality describes the situation when the common differences associated with relevant zeta-functions can be different from each other.
Theorem 4. Suppose that the set L(α, h) is linearly independent over Q, and B j , f 1 (s), f 2jl (s), K 1 , K 2jl and ϕ(s) satisfy hypotheses of Theorem 3. Then, for every ε > 0, it holds that Remark 2. As we already have noted, Theorems 3 and 4 are discrete analogues of Theorem 4.2 from [11], but here, instead of algebraic independence of the parameters α 1 , ..., α r , the linear independence of the elements of the sets L(α, h) and L(α, h) over the set of rational numbers Q are used, respectively.

A discrete limit theorem
Theorem 3 is obviously a special case of Theorem 4, so it is enough to prove Theorem 4. For this aim, we use a discrete mixed joint limit theorem (Theorem 5 below) in the sense of weakly convergent probability measures in the space of analytic functions. Since Theorem 5 is valid for more general ϕ(s) ∈ M, we will formulate the theorem in such a general setting. In this section, we first prove two auxiliary lemmas which play essential roles in the proof of Theorem 5. We introduce certain topological structure. Let γ be the unit circle on the complex plane, and Ω 1 and Ω 2 be two tori defined as where γ p = γ for all p ∈ P, and γ m = γ for all m ∈ N 0 , respectively. It is well known that both tori Ω 1 and Ω 2 are compact topological Abelian groups with respect to the product topology and pointwise multiplication. Therefore, with Ω 2j = Ω 2 for all j = 1, ..., r also is a compact topological group. This gives that we can define the probability Haar measure m H on (Ω, B(Ω)), where m H is the product of Haar measures m H1 and m H2j , j = 1, . . . , r, defined on spaces (Ω 1 , B(Ω 1 )) and (Ω 2j , B(Ω 2j )), j = 1, . . . , r, respectively. This fact allows us to have the probability space (Ω, B(Ω), m H ) (for the details, see [11]). Denote by ω 1 (p) be the projection of ω 1 ∈ Ω 1 to γ p , p ∈ P, and by ω 2j (m) the projection of ω 2j ∈ Ω 2j to γ m , m ∈ N 0 , j = 1, ..., r. The definition of ω 1 (m) for general m ∈ N 0 is given by ω 1 (m) = ω 1 (p 1 ) a1 · · · ω 1 (p q ) aq according to the decomposition m = p a1 1 · · · p aq q into prime divisors. Let ω = (ω 1 , ω 21 , ..., ω 2r ) be an element of Ω.
For A ∈ B(Ω), on (Ω, B(Ω)), define Lemma 1. Suppose that the set L(α, h) is linearly independent over Q. Then Q N converges weakly to the Haar measure m H as N → ∞.
Another important auxiliary result is related with the ergodic theory. It will be used for the identification of the explicit form of the limit measure.
Recall that the set A ∈ B(Ω) is invariant with respect to the transformation Φ α,h if the sets A and Φ α,h (A) differ from each other at most by the set of zero m H -measure, and the transformation Φ α,h is ergodic if its Borel σ-field of invariant sets consists of sets having m H -measure 0 or 1.

Lemma 2.
Suppose that the set L(α, h) is linearly independent over Q. Then the transformation Φ α,h is ergodic.
Proof. This lemma is proved by the Fourier transform method. Any character χ of the torus Ω can be written in the form for any non-trivial χ, because of the linear independence of L(α, h). Using this fact, we proceed along the standard way to prove the lemma; see, for example, Section 4 of [2]. Now we are ready for the discrete mixed joint limit theorem for the tuple of the class of zeta-functions under our investigation.
Theorem 5. Suppose that ϕ(s) ∈ M, and that the set L(α, h) is linearly independent over Q. Then P N converges weakly to P Z as N → ∞.
We prove this theorem in a way whose principle has been already welldeveloped (see, for example, Theorem 4 of [12] or Lemma 5.1 of [11] as the continuous analogue). However our present situation includes a lot of parameters which may cause a trouble for the readers, so we will explain many details even if they are now standard.
Lemma 3. Suppose that the set L(α, h) is linearly independent over Q. Then, for all n, both the measures P N,n and P N,n, ω converge weakly to the same probability measure (denote it by P n ) on (H, B(H)) as N → ∞.
Proof. This is a generalization of Lemma 1 of [12]. The following proof is similar to the argument included in several previous articles, such as [4,17]. Define h n : Ω → H by h n (ω) = Z n (s, ω, α; B). This is continuous, and Therefore, P N,n = Q N • h −1 n . This fact, Lemma 1 and Theorem 5.1 of [1] give that P N,n converges weakly to m H • h −1 n as N → ∞. Next define g n : Ω → H by g n (ω) = h n (ω · ω). Then n which converges weakly to m H • g −1 n as above. Therefore, the lemma follows, because the invariance property of the Haar measure implies In the next step of the proof, we pass from Z n (s, α; B) and Z n (s, ω, α; B) to Z(s, α; B) and Z(s, ω, α; B) by the approximation in mean, respectively. First define a metric on H. For any open region G ⊂ C, let ρ G be the standard metric on H(G) which induces the topology of uniform convergence on compact subsets (see Section 1.7 of [14]). Then, for two elements f = f 1 , f 211 , . . . , f 21l(1) , . . . , f 2r1 , . . . , f 2rl(r) , g = g 1 , g 211 , . . . , g 21l(1) , . . . , g 2r1 , . . . , g 2rl(r) of H, define the metric by Proof. This is a generalization of Lemma 2 of [12]. In view of the definition of the metric , it is sufficient to show lim n→∞ lim sup and, for almost all ω 1 , and the corresponding results for periodic Hurwitz zeta-functions.
The proof of (3.1) and (3.2) are included in the proof of Lemma 3 of [10], though the formulas themselves are not explicitly stated there. The corresponding results for periodic Hurwitz zeta-functions are Theorems 4.1 and 4.4 of [21].
In the third step, we introduce one more probability measure, for almost all ω ∈ Ω, defined by Lemma 5. Suppose that L(α, h) is as in Theorem 5. Then the measures P N and P N,ω both converge weakly to the same probability measure (denote it by P ) on (H, B(H)) as N → ∞.
Then the distribution of X N,n (s) is clearly P N,n . Let X n (s) be an H-valued random element whose distribution is P n . Then Lemma 3 implies that X N,n (s) converges to X n (s) in distribution as N → ∞.
We can show that the family {P n } is tight in a standard way (see, for example, pp. 269-270 of [8]). Therefore, by Prokhorov's theorem (see [1]), we can choose a subsequence {P n k } ⊂ {P n }, which converges weakly to a certain probability measure P on (H, B(H)) as k → ∞. That is, X n k converges to P in distribution as k → ∞.
Define another random element X N (s) = Z(s + iθ N h, α; B). Then, by Lemma 4, for any ε > 0, Therefore, we can apply Theorem 4.2 of [1] to find that X N (s) converges to P in distribution, that is, P N converges weakly to P as N → ∞. Moreover, this fact implies that P does not depend on the choice of the subsequence {P n k }. Therefore, by Theorem 1.1.9 of [14], we see that P n converges weakly to P as n → ∞.
Finally we put X N,n (s) = Z n (s + iθ N h, ω, α; B) and X N (s) = Z(s + iθ N h, ω, α; B) and argue as above. It follows that X N,n (s) → X n (s) as N → ∞ in distribution. We already mentioned above that X n (s) → P in distribution. Also, using Lemma 4, we find Therefore, again using Theorem 4.2 of [1] we obtain that P N,ω converges weakly to P as N → ∞.
The final step of the proof is to identify the measure P in Lemma 5.
Lemma 6. The probability measure P coincides with the probability measure P Z .
Proof. This is a generalization of Lemma 4 of [12]. Using Lemma 2 and the classical Birkhoff-Khintchine theorem (see [3]), we just mimic the standard argument (see, for example, the proof of Theorem 4 of [2]).
The proof of Theorem 5 is completed. From Theorem 5 we can immediately deduce simpler discrete mixed joint limit theorem when the set L(α, h) is replaced by L(α, h), that is the case h 1 = h 21 = · · · = h 2j = h. Here we give the statement of such discrete mixed joint limit theorem.

Proof of Theorems 3 and 4
For the proof of universality property in the Voronin sense, we need to construct the support of the probability measure P Z in an explicit form. Assume that ϕ(s) ∈ S, and let K 1 , K 2jl , f 1 (s), f 2jl , j = 1, ..., r, l = 1, ..., l(j), be as in our most general universality result, i.e., Theorem 4. Suppose that M > 0 is a sufficiently large number such that K 1 belongs to D M = s ∈ C : σ 0 < σ < 1, |t| < M .
Since, by the conditions (i) and (b), the function ϕ(s) has just one pole at s = 1, define D ϕ = {s ∈ C : σ > σ 0 , σ = 1}. This gives us that D M ⊂ D ϕ . Moreover we can find T > 0 such that each K 2jl , for j = 1, ..., r and l = 1, ..., l(j), is a part of D T = s ∈ C : 1 2 < σ < 1, |t| < T . Lemma 7. Suppose that the set L(α, h) is linearly independent over Q. Then the support of the measure P Z is the set S Z := S ϕ × H λ (D T ).
Proof of Theorem 4 (and hence of Theorem 3). The proof of Theorem 4 we obtain by combining results of Theorem 5, Lemma 7 and the Mergelyan theorem on the approximation of analytic functions by polynomials (see [23]). We omit the details since the argument is standard and the same as in Section 4 of [10].
Finally, we would like to mention that probably it is possible to generalize our Theorem 4 in one more direction. In this paper, we consider a wide collection of periodic Hurwitz zeta-functions ζ(s, α j ; B jl ), where we attach the collection of sequences {B jl }, 1 ≤ l ≤ l(j), to a parameter α j . However the common differences h 2j are the same for all B jl , l = 1, . . . , l(j). The new idea is to consider the situation when to each B jl the attached common difference h 2jl may be different.