JOINT DISCRETE APPROXIMATION OF ANALYTIC FUNCTIONS BY HURWITZ ZETA-FUNCTIONS

Let H(D) be the space of analytic functions on the strip ... In this paper, it is proved that there exists a closed non-empty set ...such that every collection of the functions ... is approximated by discrete shifts .., of Hurwitz zeta-functions with arbitrary parameters ...

Suppose that a = {a m : m ∈ N 0 = N ∪ {0}} be a periodic sequence of complex numbers. A generalization of the function ζ(s, α) is the periodic Hurwitz zeta-function ζ(s, α; a) = ∞ m=0 a m (m + α) s , σ > 1, which also has the meromorphic continuation to the whole complex plane.
Analytic properties of the functions ζ(s, α) and ζ(s, α; a), including the approximation of analytic functions, depend on the arithmetic nature of the parameter α. Let D = {s ∈ C : 1/2 < σ < 1}. Denote by H(D) the space of analytic functions on D endowed with the topology of uniform convergence on compacta. Approximation of all functions of the space H(D) by shifts ζ(s + iτ, α) and ζ(s + iτ, α; a), τ ∈ R, is called universality of the functions ζ(s, α) and ζ(s, α; a), respectively. More precisely, the following results are known.
Denote by K the class of compact subsets of the strip D with connected complements, and by H(K) with K ∈ K the class of continuous functions on K that are analytic in the interior of K. Let measA stand for the Lebesgue measure of a measurable set A ⊂ R. Suppose that the number α is transcendental or rational = 1 or 1/2, and K ∈ K, f (s) ∈ H(K). Then, for every ε > 0, Different proofs of the latter inequality are given in [1,10,36] and [28]. The above theorem is of continuous type. Also, a similar result of discrete type is known. Denote by #A the cardinality of a set A, and let N run over the set N 0 . For α rational = 1 or 1/2, let h > 0 be arbitrary, while, for transcendental α, let h be such that exp{(2πl)/h} is irrational for all l ∈ N.
Universality of the functions ζ(s, α) and ζ(s, α; a) with algebraic irrational parameter α is a very complicated and open problem.
In [13,15], for universality of ζ(s, α), the linear independence over the field of rational numbers for the sets {log(m + α) : m ∈ N 0 } and {(log(m + α) : m ∈ N 0 ) , 2π/h} was required. This requirement is weaker than the transcendence of α, however, examples of such α are not known. In the joint case, the above sets were generalized [13,16] by There are known several results of approximation of analytic functions by shifts of the functions ζ(s, α) and ζ(s, α; a) with algebraic irrational parameter α, however, the set of approximated functions is not identified. The first results of such a kind has been obtained in [2]. Suppose that 0 < α < 1 is arbitrary. Then there exists a closed non-empty subset F α ⊂ H(D) such that, for every compact K ⊂ D, f (s) ∈ F α and ε > 0, inequality (1.1) holds. The analogical statements for the functions ζ(s, α; a) and the Lerch zeta-function are given in [9] and [21], respectively. Generalizations of [2] for the Mishou theorem were obtained in [24]. In [8], the following joint approximation theorem for Hurwitz zeta-functions has been proved. Theorem 1. Suppose that the numbers 0 < α j < 1, α j = 1/2, j = 1, . . . , r, are arbitrary. Then there exists a closed non-empty set F α1,...,αr ⊂ H r (D) such that, for every compact sets K 1 , . . . , K r ⊂ D, (f 1 , . . . , f r ) ∈ F α1,...,αr and ε > 0, Moreover, the limit exists for all but at most countably many ε > 0.
It will be proved that the set F α,h is the support of a certain H r (D)-valued random element.

Probabilistic results
Denote by B(X) the Borel σ-field of the space X, and, for A ∈ B(H r (D)), define In this section, we deal with weak convergence of P N,α,h as N → ∞.
We start with definition of one probability space. Define where γ m = {s ∈ C : |s| = 1} for all m ∈ N 0 . By the Tikhonov theorem, with the product topology and pointwise multiplication, the torus Ω is a compact topological Abelian group. Therefore, Ω r = Ω 1 × · · · × Ω r , where Ω j = Ω for all j = 1, . . . , r, again is a compact topological Abelian group. Thus, on (Ω r , B(Ω r )), the probability Haar measure m H can be defined, and we have the probability space (Ω r , B(Ω r ), m H ). Denote by ω j (m) the mth component of an element ω j ∈ Ω j , j = 1, . . . , r, m ∈ N. Characters of the group Ω r are of the form where the sign " * " shows that only a finite number of integers k jm are distinct from zero. Therefore, putting k = {k jm : k jm ∈ Z, m ∈ N 0 }, j =, . . . , r, we have that the Fourier transform g(k 1 , . . . , k r ) of a probability measure µ on (Ω r , B(Ω r )) is given by Let Q α,h be the probability measure on (Ω r , B(Ω r )) having the Fourier transform Proof. In view of (2.1), the Fourier transform g N,α,h (k 1 , . . . , k r ) of Q N,α,h is given by Thus, g N,α,h (k 1 , . . . , k r ) = 1 for (k 1 , . . . , k r ) ∈ A(α, h). If (k 1 , . . . , k r ) ∈ B(α, h), then by the sum formula of geometric progression, we have Define ζ n (s, α) = (ζ n (s, α 1 ), . . . , ζ n (s, α r )), where In view of the definition v n (m, α j ), the latter Dirichlet series are absolutely convergent for σ > 1/2. For A ∈ B(H r (D)), define To obtain the weak convergence for V N,n,α,h as N → ∞, introduce the mapping u n,α : Ω r → H r (D) given by where ζ n (s, α, ω) = (ζ n (s, α 1 , ω 1 ), . . . , ζ n (s, α r , ω r )) with Obviously, the latter series also are absolutely convergent for σ > 1/2. Therefore, the mapping u n is continuous, hence, it is (B(Ω r ), B(H r (D)))-measurable. Thus, the measure Q α,h defines the unique probability measure V α,h on (H r (D), B(H r (D))) by the formula Moreover, the definitions of V N,n,α,h and Q N,α,h imply the equality All these remarks together with Lemma 1 and the property of preservation of weak convergence under continuous mappings lead to the following limit lemma.
To obtain a limit theorem for P N,α,h , we need the estimation a distance between ζ n (s, α) and ζ(s, α). Let g 1 , g 2 ∈ H(D). Recall that ρ(g 1 , g 2 ) = ∞ l=1 2 −l sup s∈K l |g 1 (s) − g 2 (s)| 1 + sup s∈K l |g 1 (s) − g 2 (s)| , where {K l : l ∈ N} is a certain sequence of compact subsets of the strip D, is a metric on H(D) inducing its topology of uniform convergence on compacta. Let g 1 = (g 11 , . . . , g 1r ), g 2 = (g 21 , . . . , g 2r ) ∈ H r (D). Then is a metric on H r (D) that induces the product topology. Let θ be the same parameter as in definition of v n (m, α j ), and where Γ (s) is the Euler gamma-function. Then the following integral representation is known [28].
We will use some mean square results of discrete type. For the proof of them, the next lemma connecting the continuous and discrete mean squares is useful. Lemma 5. Suppose that 0 < α 1, 1/2 < σ < 1 and h > 0 are fixed numbers. Then, for every t ∈ R, Proof. It is well known that Therefore, an application of Lemma 4 with δ = h gives the estimate of the lemma.
The next lemma is very important for the proof of weak convergence for P N,α,h . Thus, let K ⊂ D be an arbitrary compact set. There exists ε > 0 such that all points of the set K lie in the strip {s ∈ C : 1/2 + 2ε σ 1 − ε}. Let s = σ + it ∈ K, and θ 1 = σ − 1/2 − ε > 0. Then, in view of Lemma 3 and the residue theorem, where R n (s, α) = Res z=1 ζ(s + z, α)l n (z, α) Hence, for s ∈ K, Therefore, where The crucial role in the estimation of l n (s, α) is played by the gamma-function. It is well known that there exists c > 0 such that, uniformly in σ 1 σ σ 2 , This estimate leads, for σ + it ∈ K, to Therefore, in view of Lemma 5, By estimate (2.3) again, we find that, for s ∈ K, Therefore, This, together with (2.4) and (2.2) proves the lemma. Now, we define the marginal measures of V n,α,h . For A ∈ B(Ω j ), j = 1, . . . , r define Then by Lemma 1 of [27], Q N,αj ,hj converges weakly to a certain probability measure Q αj ,hj on (Ω j , B(Ω j )) as N → ∞, j = 1, . . . , r. Let the mapping u n,αj : Ω j → H(D) be given by u n,αj (ω j ) = ζ n (s, α j , ω j ). Define Then in [27,Lemma 4], the following statement has been obtained.
We apply Lemma 7 for the family of probability measures {V n,α,h : n ∈ N}. Lemma 8. The family {V n,α,h : n ∈ N} is tight.
Proof. Let ε > 0 be an arbitrary number. By Lemma 7, there exist compact sets K 1 , . . . , K r ⊂ H(D) such that for all n ∈ N. Let K = K 1 × · · · × K r . Then K is a compact set in H r (D). Denoting Now we are in position to prove a limit theorem for P N,α,h .
Theorem 3. On (H r (D), B(H r (D))), there exists a probability measure P α,h such that P N,α,h converges weakly to P α,h as N → ∞.
Moreover, let Y n,α,h be the H r (D)-valued random element having the distribution V n,α,h . Then, in view of Lemma 2, where D − → means the convergence in distribution.
This, (2.6) and (2.7) together with Theorem 4.2 of [4] show that Since the latter relation is equivalent to weak convergence of P N,α,h to P α,h as N → ∞, the theorem is proved.

Proof of approximation
Denote by F α,h the support of the limit measure P α,h in Theorem 3. Thus F α,h ⊂ H r (D) is a minimal closed set such that P α,h (F α,h ) = 1. The set F α,h consists of all elements g ∈ H r (D) such that, for every open neighbourhood G of g, the equality P α,h (G) > 1 is satisfied. Obviously, F α,h = ∅.
1. Let (f 1 (s), . . . , f r (s)) ∈ F α,h . Define the set Therefore, these boundaries do not intersect for different ε. Hence, the set G ε is a continuity set of the measure P α,h for all but at most countably many ε > 0. Therefore, Theorem 3 together with equivalent of weak convergence of probability measures in terms of continuity sets implies that lim N →∞ P N,α,h (G ε ) = P α,h (G ε ) > 0 for all but at most countably many ε > 0, and the second assertion of the theorem is proved.