Joint universality of periodic zeta-functions with multiplicative coefficients

which has meromorphic continuation to the whole complex plane with unique simple pole at the point s = 1 with residue 1. Let D = {s ∈ C: 1/2 < σ < 1}. Voronin considered approximation of analytic functions defined on D by shifts ζ(s+ iτ), τ ∈ R. For the last version of the Voronin universality theorem, it is convenient to use the following notation. Denote by K the class of compact subsets of the strip D with connected complements,

The above shifts are very simple, τ 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 Pańkowski [31] using the shifts ζ(s + iϕ(τ )) and ζ(s + iϕ(k)) 1 T log T was introduced. This conjecture is inspired by the Montgomery pair correlation conjecture [28] that where α 1 < α 2 are arbitrary real numbers, and δ(α 1 , α 2 ) = 1 if 0 ∈ [α 1 , α 2 ], 0 otherwise. Now we will state a joint universality theorem for Dirichlet L-functions involving the sequence {γ k } obtained in [18]. Denote by K the class of compact subsets of the strip D with connected complements, and by H 0 (K) with K ∈ K the class of continuous nonvanishing functions on K that are analytic in the interior of K.
Here #A denotes the cardinality of the set A, and N runs over the set N. Now we recall the definition of the periodic zeta-function, which is an object of investigation of the present note. Let a = {a m : m ∈ N} be a periodic sequence of complex numbers with minimal period q ∈ N. Then the periodic zeta-function ζ(s; a) is defined, for σ > 1, by the Dirichlet series a m m s and has an analytic continuation to the whole complex plane, except for a simple pole at the point s = 1 with residue The sequence a is called multiplicative if a 1 = 1 and a mn = a m a n for all coprimes m, n ∈ N. If 0 < α 1 is a fixed number, then the function ζ(s, α; a) = ∞ m=0 a m (m + α) s , σ > 1, and its meromorphic continuation are called the periodic Hurwitz zeta-function. In [15] and [3], under hypothesis (1), joint universality theorems involving sequence {γ k } for the pair consisting from the Riemann and Hurwitz zeta-functions and their periodic analogues, respectively, were obtained, while in [23], such theorems were proved for Hurwitz zeta-functions.
For j = 1, . . . , r, let a j = {a jm : m ∈ N} be a periodic sequences of complex numbers with minimal period q j ∈ N, and let ζ(s; a j ) be the corresponding zeta-function. The main result of the paper is the following theorem.
Theorem 2. Suppose that the sequences a 1 , . . . , a r are multiplicative, h 1 , . . . , h r are positive algebraic numbers linearly independent over the field of rational numbers, and estimate (1) is true. For j = 1, . . . , r, let K j ∈ K and f j (s) ∈ H 0 (K j ). 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.
it follows that ζ(−2m) = 0 for all m ∈ N, and the zeros s = −2m of ζ(s) are called trivial. Moreover, it is known that ζ(s) has infinitely many of so-called complex nontrivial zeros ρ k = β k + iγ k lying in the strip {s ∈ C: 0 < σ < 1}. The famous Riemann hypothesis, one of seven Millennium problems, asserts that β k = 1/2, i.e., all nontrivial zeros lie on the critical line σ = 1/2. There exists a conjecture that all nontrivial zeros of ζ(s) are simple. We recall some properties of the sequence By the definition, a sequence {x k : 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 k } is distributed irregularly, the following statement is true for it.
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].
Obviously, the uniform distribution modulo 1 of the sequence shows its nonlinear character.
The following statement is well known; see, for example, [34].

Limit theorems
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 σ-field of the space X, and P denote the set of all prime numbers. Define Then Ω is a compact topological Abelian group. Moreover, let where Ω j = Ω for j = 1, . . . , r. Then again Ω r is a compact topological Abelian group. Therefore, on (Ω r , B(Ω r )), the probability Haar measure m r H can be defined. This gives the probability space (Ω r , B(Ω r ), m r H ). Denote by ω(p) the pth component, p ∈ P, of an element ω j ∈ Ω j , j = 1, . . . , r. For brevity, let ω = (ω 1 , . . . , ω r ) ∈ Ω r , ω 1 ∈ Ω 1 , . . . , ω r ∈ Ω r , a = (a 1 , . . . , a r ), and on the probability space Note that the latter products, for almost all ω j , are uniformly convergent on compact subsets of the strip D. Since the periodic sequences a j , j = 1, . . . , r, are bounded, the proofs of the above assertions completely coincides with those of Lemma 5.1.6 and Theorem 5.1.7 from [13]. More general results are given in [1]. Denote by P ζ the distribution of the random element ζ(s, ω; a), i.e., Put h = (h 1 , . . . , h r ), and, for A ∈ B(H r (D)), define where ζ(s; a) = ζ(s; a 1 ), . . . , ζ(s; a r ) .
In this section, we will prove the following limit theorem.
We start the proof of Theorem 3, as usual, with a limit lemma in the space Ω r . In this lemma, the uniform distribution modulo 1 of the sequence {γ k a}, a ∈ R \ {0}, and the property of the numbers h 1 , . . . , h r essentially are applied.
For A ∈ B(Ω r ), define Before the statement of a limit theorem for Q N , we recall one result of Diophantine type.
Lemma 4. Suppose that λ 1 , . . . , λ r ∈ C are algebraic numbers such that the logarithms log λ 1 , . . . , log λ r are linearly independent over Q. Then, for any algebraic numbers β 0 , . . . , β r , not all zero, we have where H is the maximum of the heights of β 0 , β 1 , . . . , β r , and C is an effectively computable number depending on r and the maximum of the degrees of β 0 , β 1 , . . . , β r .
The lemma is the well-known Baker theorem on logarithm forms; see, for example [2]. where the star " * " shows that only a finite number of integers k jp are distinct from zero. Therefore, the Fourier transform of Q N is where k j = (k jp : k jp ∈ Z, p ∈ P), j = 1, . . . , r. Thus, by the definition of Q N , Obviously, g N (0, . . . , 0) = 1.
Now, suppose that k = (0, . . . , 0). Then there exists j ∈ {1, . . . , r} such that k j = 0. Thus, there exists a prime number p such that k jp = 0. Define Then, in view of a property of the numbers h 1 , . . . , h r , we have a p = 0. The numbers a p are algebraic, and the set {log p: p ∈ P} is linearly independent over Q. Therefore, by Lemma 4, Then the latter series are absolutely convergent for σ > 1/2. Actually, since v n (m) m −L/n θ with every L > 0, the latter series are absolutely convergent even in the whole complex plane. For B(H r (D)), define where ζ n (s; a) = ζ n (s; a 1 ), . . . , ζ n (s; a r ) .

Moreover, let
ζ n (s, ω j ; a j ) = . Proof. Since the series for ζ n (s, ω j ; a j ) are absolutely convergent for σ > 1/2, the function u n is continuous, hence (B(Ω r ), B(H r (D)))-measurable. Therefore, the measure V n is defined correctly. The definitions of Q N , V N,n and u n imply the equality V N,n = Q N u −1 n . Therefore, the lemma follows from Lemma 5 and a preservation of weak convergence under continuous mappings; see [4, Thm. 5.1].
The limit measure V n in Lemma 6 is independent on h and {γ k } and has a good convergence property, which is the next lemma.
Lemma 7. Suppose that the sequences a 1 , . . . , a r are multiplicative. Then V n converges weakly to P ζ as n → ∞.
Proof. In [17], the weak convergence for was considered, and it was obtained its weak convergence to P ζ as T → ∞, and that V n also converges weakly to P ζ as n → ∞. In other words, V n andP T have the same limit measure P ζ .
In view of Lemma 7, to prove Theorem 3, it suffices to show that P N , as N → ∞, and V n , as n → ∞, have a common limit measure. For this, a certain closeness of ζ(s; a) and ζ n (s; a) is needed.
There exists a sequence {K l : l ∈ N} ⊂ D of compact subsets such that K l ⊂ K l+1 , for all l ∈ N, and if K ⊂ D is a compact set, then K ⊂ K l for some l. Then, putting, for g 1 , g 2 ∈ H(D), we have a metric in H(D) inducing its topology of uniform convergence on compacta. Hence, ρ(g 1 , g 2 ) = max 1 j r ρ(g 1j , g 2j ), . . , g 1r ), g 2 = (g 21 , . . . , g 2r ) ∈ H r (D), is a metric in H r (D) inducing its product topology. Note that, in the proof of the next lemma, the multiplicativity of the sequences a j , j = 1, . . . , r, is not used.
Proof. By the definitions of the metrics ρ and ρ, it is sufficient to show that, for every compact set K ⊂ D, sup s∈K ζ(s + ih j γ k ; a j ) − ζ n (s + ih j γ k ; a j ) = 0, 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 a be arbitrary. We consider ζ(s + ihγ k ; a) and ζ n (s + ihγ k ; a). Let θ be as in the definition of v n (m). Then the representation where R n (s; a) = al n (1 − s) 1 − s , and a is the residue of ζ(s; a) at the point s = 1. Let K ⊂ D be an arbitrary compact set, and ε > 0 be such that 1/2 + 2ε σ 1 − ε for s ∈ K. Then, in view of (6), for s = σ + iv ∈ K, Hence, taking t in place of t + v and θ 1 = σ − ε − 1/2, we have where Estimate (1) is applied for estimation of the first factor of the integrated function in the integral I. It is well known that, for τ ∈ R, The same estimate is also true for the derivative of ζ(s; a). Let δ = ch(log γ N ) −1 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 l n (s) show that I ε,h,K n −ε and Z h,K n 1/2−2ε log N N .
Proof of Theorem 3. We will use the random element language. Denote by X n = X n (s) the H r (D)-valued random element having the distribution V n , where V n is the limit measure in Lemma 6. Then, by Lemma 7, where D → means the convergence in distribution. Now, let the random variable η N be defined on a certain probability space with a measure µ, and Define the H r (D)-valued random element X N,n = X N,n (s) = ζ n (s + ihη N ; a).

Proof of Theorem 2
We start with the explicit form of the support of the measure P ζ . Recall that the support of a probability measure P is a minimal closed set S P such that P (S P ) = 1.
Lemma 9. The support of the measure P ζ is the set S r .
Proof. The space H r (D) is separable. Therefore [4], From this it follows that it suffices to consider the measure P ζ on the rectangular sets A = A 1 × · · · × A r , A 1 , . . . , A r ∈ B(H(D)).
It is known [19] that the support of P ζ j (A j ) = m jH ω j ∈ Ω j : ζ(s, ω j ; a j ) ∈ A j , j = 1, . . . r, 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 p 1 (s), . . . , p r (s) such that In view of Lemma 9, the set G ε = (g 1 , . . . , g r ) ∈ H r (D): sup is an open neighbourhood of an element of the support of the measure P ζ . Hence, Therefore, by Theorem 3 and the equivalent of weak convergence of probability measures in terms of open sets, lim inf N →∞ This, the definitions of P N and G ε , 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 G ε = (g 1 , . . . , g r ) ∈ H r (D): sup ThenĜ ε is a continuity set of the measure P ζ for all but at most countably many ε > 0, moreover, in view of (12), the inclusion G ε ⊂Ĝ ε is valid. Therefore, Theorem 3, the equivalent of weak convergence of probability measures in terms of continuity sets and (13) lead the inequality lim N →∞ P N (Ĝ ε ) = P ζ (Ĝ ε ) > 0 for all but at most countably many ε > 0. This, the definitions of P N andĜ ε prove the second part of the theorem.