Joint Universality of the Zeta-Functions of Cusp Forms

: Let F be the normalized Hecke-eigen cusp form for the full modular group and ζ ( s , F ) be the corresponding zeta-function. In the paper, the joint universality theorem on the approximation of a collection of analytic functions by shifts ( ζ ( s + ih 1 τ , F ) , . . . , ζ ( s + ih r τ , F )) is proved. Here, h 1 , . . . , h r are algebraic numbers linearly independent over the ﬁeld of rational numbers.


Introduction
The series of the types In [1], Voronin discovered a very interesting and important property of ζ(s) to approximate a wide class of analytic functions by shifts ζ(s + iτ), τ ∈ R, and called it universality. Later, it turned out that some other zeta-functions also are universal in the Voronin sense. This paper is devoted to the universality of zeta-functions of certain cusp forms. Let SL(2, Z) = a b c d : a, b, c, d ∈ Z, ad − bc = 1 be the full modular group. If the function F(z) is holomorphic in the upper half-plane Imz > 0, and for all elements of SL(2, Z) with some κ ∈ 2N satisfies the functional equation where F(z) is called a modular form of weight κ for the full modular group. Then, F(z) has Fourier series expansion In this case, c(1) = 0; therefore, the form F(z) can be normalized, and thus, we may suppose that c(1) = 1. Now, we suppose that F(z) is a normalized Hecke-eigen cusp form of weight κ for the full modular group. Then, the zeta-function ζ(s, F) can be written, for σ > κ+1 2 , as a product over primes where α(p) and β(p) are conjugate complex numbers satisfying the equality α(p) + β(p) = c(p). In the paper [2], the universality of the function ζ(s, F) was proved. Let D κ = s ∈ C : κ 2 < σ < κ+1 2 , K F be the class of compact subsets of the strip D κ with connected complements, and H 0,F (K), K ∈ K F the class of continuous nonvanishing functions on K that are analytic in the interior of K. Moreover, let measA denote the Lebesgue measure of a measurable set A ⊂ R. Then, in [2], the following theorem was obtained. Theorem 1. Suppose that K ∈ K F and f (s) ∈ H 0,F (K). Then, for every ε > 0, Theorem 1 shows that there are infinitely many shifts ζ(s + iτ, F) approximating a given function f (s) ∈ H 0,F . In the shifts ζ(s + iτ, F) of Theorem 1, τ takes arbitrary real values; therefore, the theorem is of continuous type. Further, discrete universality theorems for the function ζ(s, F) are known. In [3,4], the discrete universality theorems with shifts ζ(s + ikh, F), k ∈ N, h > 0 being a fixed number, were proved. Denote by H(D κ ) the space of analytic on D κ functions endowed with the topology of uniform convergence on compacta. The paper [5] is devoted to the universality for compositions Φ(ζ(s, F)) with certain operators Φ : H(D κ ) → H(D κ ). The results of the latter paper were applied in [6] for the functional independence of the compositions Φ(ζ(s, F)).
Let, for a fixed l ∈ N, denote the Hecke subgroup of the group SL(2, Z). If F(z) satisfies (1) for all elements of Γ 0 (l), then F(z) is called a cusp form of weight κ and level l. The form F(z) is called a new form if it is not a cusp form of level l 1 | l. In [7], a universality theorem was obtained for zeta-functions of new forms. The universality theorem of [2] was generalized in [8] for shifts ζ(s + iϕ(τ), F) with differentiable function ϕ(τ) satisfying the estimates (ϕ (τ)) −1 = o(τ) and ϕ(2τ) max τ t 2τ (ϕ (t)) −1 τ as τ → ∞. The discrete version of results of [8] is given in [9]. In [10], the shifts ζ(s + iγ k , F), where {γ k : k ∈ N} is the sequence of nontrivial zeros of ζ(s), are used.
The joint universality of zeta-and L-functions is a more complicated problem of analytic number theory. In this case, a collection of analytic functions are simultaneously approximated by a collection of shifts of zeta-functions. The first result in this direction also belongs to Voronin. He considered [11] the functional independence of Dirichlet L-functions L(s, χ) with pairwise nonequivalent Dirichlet characters χ and, for this, he obtained their joint universality. The paper [12] is devoted to the joint universality for zeta-functions of new forms twisted by Dirichlet characters, i.e., for the functions with pairwise nonequivalent Dirichlet characters χ 1 , . . . , χ r . Joint universality theorems with generalized shifts ζ(s + iϕ j (k), F), j = 1, . . . , r, with some differentiable functions ϕ j (τ) can be found in [13]. Continuous and discrete joint universality theorems for more general zeta-functions are given in [14][15][16].
Our aim is to obtain a joint universality theorem for zeta-functions of normalized Hecke-eigen cusp forms by using different shifts. The first of the denseness results for shifts of a universal function were discussed in [17].
The main result of the paper is the following statement.

Theorem 2.
Suppose that h 1 , . . . , h r are real algebraic numbers linearly independent over the field of rational numbers Q. For j = 1, . . . , r, let K j ∈ K F and f j (s) Moreover "lim inf" can be replaced by "lim" for all but at most countably many ε > 0.
For the proof of Theorem 2, we will apply the probabilistic approach based on a limit theorem in the space of analytic functions.

Mean Square Estimates
Recall the metric in the space H(D κ ). Let {K l : l ∈ N} ⊂ D κ be a sequence of compact subsets such that is a compact, then K ⊂ K l for some l. For example, we can take K l closed rectangles. Then For g j = (g j1 , . . . , g jr ) ∈ H r (D κ ), j = 1, 2, define Then, ρ is a metric in H r (D κ ) inducing the product topology. Let θ > 1 2 be a fixed number, and Then, the series is absolutely convergent in every fixed half plane σ >σ. However, for our aim, this convergence is sufficient only for σ > κ 2 . For brevity, let h = (h 1 , . . . , h r ), and ζ n (s + ihτ, F) = (ζ n (s + ih 1 τ, F), . . . , ζ n (s + ih r τ, F)).

Lemma 1. For all h,
Proof. By the definitions of the metrics ρ and ρ, it suffices to show that, for every h ∈ R and compact set K ⊂ D κ , It is well known that for fixed κ where σ means that the implied constant depends on σ. Therefore, Let where Γ(z) denotes the Euler gamma-function and θ is a number from the definition of v n (m). Using the Mellin formula Therefore, in virtue of the definition of the function v n (m), we obtain that, for σ > κ 2 , Let K ∈ D κ be a fixed compact set. Then, there exists ε > 0 such that, for all s = σ + it ∈ K, the inequalities κ 2 + 2ε < σ < κ+1 2 − ε are satisfied. We take, for such σ, Then, θ 1 < 0. Therefore, by the residue theorem and (3), Hence, for all s ∈ K, Thus, in view of (2), Here, we used the estimate Estimate (4) proves the lemma.
Then, repeating the proof of Lemma 1 and using estimate (5), we arrive to the following statement.

Limit Theorems
On and denote by P ζ,F its distribution, i.e.,

Theorem 3.
Suppose that h 1 , . . . , h r are real algebraic numbers linearly independent over Q, and Then, P T,F converges weakly to P ζ,F as T → ∞.
We divide the proof of Theorem 3 into several lemmas.
The lemma is a Baker result on linear forms of logarithm; see, for example, ref. [19]. For A ∈ B(Ω), define Proof. We apply the Fourier transform method. Denote by g T (k 1 , . . . , k r ), k j = {k pj : k pj ∈ Z, p ∈ P}, j = 1, . . . , r the Fourier transform of Q T . By the definition of Q T , we have where the star shows that only a finite number of integers k pj are not zero. Obviously, g T (0, . . . , 0) = 1.
Now, suppose that (k 1 , . . . , k r ) = (0, . . . , 0). Then, there exists a prime number p such that k pj = 0 for some j. Therefore, h j k pj = 0 because the numbers h 1 , . . . , h r are linearly independent over Q. Thus, in view of Lemma 3, This and (6) imply Therefore, by (7), For A ∈ B(H r (D κ )), define Moreover, let the mapping u n : Ω → H r (D κ ) be given by Since the series for ζ n (s, ω j , F) are absolutely convergent for σ > κ 2 , the mapping u n,F is continuous. Moreover, by the definitions of Q T and P T,n,F , we have P T,n,F = Q T u −1 n,F . This equality, continuity of u n,F , Lemma 4, the well-known properties of weak convergence, and the invariance of the Haar measure m H lead to the following lemma.

Lemma 5.
Let h 1 , . . . , h r be the same as Theorem 3. Then, P T,n,F and P T,n,Ω,F both converge weakly to the measure V n,F as T → ∞.
Additionally to P T,F , define

Lemma 6.
Let h 1 , . . . , h r be the same as Theorem 3. Then, on (H r (D κ ), B(H r (D κ ))), there exists a probability measure P F such that P T,F and P T,Ω,F both converge weakly to P F as T → ∞.
Proof. Since the series for ζ n (s, F) is absolutely convergent, by a standard way it followssee, for example [14,18]-that the sequence {V n,F : m ∈ N} is tight, i.e., for every ε > 0, there exists a compact set K ⊂ H r (D κ )) such that V n,F (K) > 1 − ε for all n ∈ N. Hence, by the Prokhorov theorem, see [20], the sequence {V n,F } is relatively compact, i.e., each of its subsequences contains a subsequence {V n k ,F } such that V n k ,F converges weakly to a certain probability measure P F on (H r (D κ ), B(H r (D κ ))) as k → ∞.
Let ξ T be a random variable defined on a certain probability space with measure ν and uniformly distributed on [0, T]. Define the H r (D κ )-valued random element X T,n,F = X T,n,F (s) = ζ n (s + ihξ T , F) and denote by X n,F = X n,F (s) the H r (D κ )-valued random element having the distribution V n,F . Then, by Lemma 5, we have where D − −− → T→∞ means the convergence in distribution. Moreover, since V n k ,F converges weakly to P F , the relation is true. Let Then, using Lemma 1, we find that for every ε > 0, The later equality together with (8) and (9), and Theorem 4.2 of [20] lead to the relation This proves that P T,F converges weakly to P F as T → ∞.
Then, repeating the above arguments using Lemmas 2 and 5, and relation (11), we obtain that X T,n,F D − −− → T→∞ P F , and this is equivalent to weak convergence of P T,Ω,F to P F as T → ∞. The lemma is proved.
To prove Theorem 3, it remains to show that P F = P ζ,F . For this, we will apply some elements of the ergodic theory. For brevity, let h τ = p −ih 1 τ : p ∈ P , . . . , p −ih r τ : p ∈ P , τ ∈ R.
Since the Haar measure m H is invariant, the transformation ϕ τ is measure-preserving and {ϕ τ : τ ∈ R} is a one-parameter group. A set A ∈ B(Ω) is called invariant with respect to the group {ϕ τ } if the sets A and ϕ τ (A), τ ∈ R, differ one from another at most by a set of m H -measure zero.

Lemma 7.
Let h 1 , . . . , h r be the same as Theorem 3. Then, the group {ϕ τ } is ergodic, i.e., the σ-field of invariant sets consists of sets having m H -measure 1 or 0.
Proof. The characters χ of the group Ω are of the form This fact already was used in the proof of Lemma 4. Let A be an arbitrary invariant set, I A its indicator function, and χ be a nontrivial character. Preserving the notation of the proof of Lemma 4, we have (k 1 , . . . , k r ) = (0, . . . , 0) and B k 1 ,...,k r = 0. Therefore, there exists τ 0 ∈ R such that χ(h τ ) = exp −iτ 0 B k 1 ,...,k r = 1.
Moreover, in view of the invariance of A, we have for almost all ω ∈ Ω. Denote byÎ A the Fourier transform of I A . Then, by (13), This and (12) show thatÎ Now, let χ 0 denote the trivial character of Ω, and suppose thatÎ A (χ 0 ) = α. Then, in view of (14), we find thatÎ Hence, I A (ω) = α for almost all ω ∈ Ω. Since I A is the indicator function, I A (ω) = 1 or I A (ω) = 0 for almost all ω. Thus, m H (A) = 1 or m H (A) = 0, and the lemma is proved.
Proof of Theorem 3. We have mentioned that it suffices to show that P F = P ζ,F . By Lemma 6 and the equivalent of weak convergence in terms of continuity sets, we have lim T→∞ P T,Ω,F (A) = P F (A) (15) for a continuity set A of the measure P F , i.e., P F (∂A) = 0, where ∂A is the boundary of A.
Therefore, by (15), we obtain that P F (A) = P ζ,F (A) for all continuity sets A of P F (A). Hence, P F = P ζ,F , and the theorem is proved.

Proof of Theorem 2
Recall that the support of the measure P ζ,F is a minimal closed set S F ⊂ H r (D κ ) such that P ζ,F (S F ) = 1.
Let ζ(s, ω, F) be the H(D κ )-valued random element defined on the probability space (Ω, B(Ω), m H ), where m H is the Haar measure. Then, it is known [10] that the support of the distribution of ζ(s, ω, F) is the set {g ∈ H(D κ ) : g(s) = 0 or g(s) ≡ 0}. Thus, the same set is the support of the distributions of ζ(s, ω j , F), j = 1, . . . , r. Since the measure m H is the product of the measures m jH , j = 1, . . . , r, we have m H ω ∈ Ω : ζ(s, ω, F) ∈ A = r ∏ j=1 m jH ω j ∈ Ω j : ζ(s, ω j , F) ∈ A j . This equality, the minimality of the support, and the support of the distributions of ζ(s, ω j , F) prove the lemma.
Proof of Theorem 2. By the Mergelyan theorem on the approximation of analytic functions by polynomials [22], there exist polynomials p 1 (s), . . . , p r (s) such that sup 1≤j≤r sup s∈K j f j (s) − e p j (s) < ε 2 .
In view of Lemma 8, the set G ε is an open neighborhood of an element (e p 1 (s) , . . . , e p r (s) ) in support of the measure P ζ,F . Hence, This, Theorem 3 and the equivalent of weak convergence in terms of open sets, and the definitions of P T,F and G ε prove the theorem with "lim inf". Define one more set G ε = (g 1 , . . . , g r ) ∈ H r (D κ ) : sup 1≤j≤r sup s∈K j g j (s) − f j (s) < ε , There ∂Ĝ ε 1 ∂Ĝ ε 2 = ∅ for ε 1 = ε 2 . This shows that P ζ,F ∂Ĝ ε = 0 for all but, for those countable, many ε > 0. Moreover, (17) and (18) imply that P ζ,F Ĝ ε > 0. This, Theorem 3 and the equivalent of weak convergence of probability measures in terms of continuity sets, and the definitions of P T,F andĜ ε prove the theorem with "lim".