Extension of the discrete universality theorem for zeta-functions of certain cusp forms ∗

In [18], S.M. Voronin discovered the universality property of the Riemann zeta-function ζ(s), s = σ + it, on the approximation of a wide class of analytic functions by shifts ζ(s + iτ), τ ∈ R. Later, it turned out that some other zeta and L-functions also are universal in the Voronin sense, among them, zeta-functions of certain cusp forms. We recall their definition. Let SL(2,Z) def = {( a b c d ) : a, b, c, d ∈ Z, ad− bc = 1 }


Introduction
In [18], S.M. Voronin discovered the universality property of the Riemann zeta-function ζ(s), s = σ + it, on the approximation of a wide class of analytic functions by shifts ζ(s + iτ ), τ ∈ R. Later, it turned out that some other zeta and L-functions also are universal in the Voronin sense, among them, zeta-functions of certain cusp forms.We recall their definition.We assume additionally that the cusp form F (z) is a normalized Hecke-eigen cusp form, i.e., is an eigen form of all Hecke operators Then it is known that the Fourier coefficients c(m) = 0. Therefore, after normalization, we can assume that c(1) = 1.
The zeta-function ζ(s, F ) associated to a normalized Hecke-eigen cusp form F (z) of weight κ is defined, for σ > (κ + 1/2, by the Dirichlet series and can be analytically continued to an entire function.Moreover, as the Riemann zetafunction, the function ζ(s, F ), for σ > (κ + 1)/2, has the Euler product expansion over primes ζ(s, F ) = The universality of ζ(s, F ) was obtained in [7].Let D F = {s ∈ C: κ/2 < σ < (κ + 1)/2}.Denote by K F the class of compact subsets of the strip D F with connected complements and by H 0 (K), K ∈ K F , the class of continuous non-vanishing functions on K that are analytic in the interior of K. Let meas A stand for the Lebesgue measure of a measurable set A ⊂ R. Then the main theorem of [7] is of the following form.
The discrete version of universality for zeta-functions was proposed by A. Reich.In [16], he obtained a discrete universality theorem for Dedekind zeta-functions.In his theorem, τ takes values from the arithmetic progression {kh: k ∈ N 0 = N ∪ {0}}, where h > 0 is a fixed number.The first discrete universality theorem for ζ(s, F ) attached to a new form F (z), under a certain arithmetical hypothesis for the number h, was proved in [9].In [10], this hypothesis was removed, and the following statement was obtained.
Theorem 2. Let #A denote the cardinality of a set A. Suppose that K ∈ K F , f (s) ∈ H 0 (K), and h > 0 is an arbitrary fixed number.Then, for every ε > 0, lim inf There exists a problem to prove analogues of Theorem 2 for the sets different from the progression {kh: k ∈ N 0 }.The first attempt in this direction, in the case of the Riemann zeta-function, was made in [2], where the arithmetical progression was replaced by the set {k α h: k ∈ N 0 } with a fixed α, 0 < α < 1.An analogue of the theorem from [2] for the function ζ(s, F ) was given in [5].Ł. Pańkowski investigating the joint universality of Dirichlet L-functions extended [15] the theorem of [2] for all non-integers α > 0 and more general sets of the type {hk α log β k}, where The aim of this paper is to prove a discrete universality theorem for the function ζ(s, F ) when τ in ζ(s + iτ, F ) runs over some general sequence of real numbers.
For the definition of a class of sequences for τ , we will use the notion of uniform distribution modulo 1.Let {u} denote the fractional part of u ∈ R, and let χ I be the indicator function of the set I. We remind that a sequence Let k 0 ∈ N. We say that a function ϕ ∈ U (k 0 ) if the following hypotheses are satisfied: For example, the function ϕ(t) = t log α t with 0 < α < 1 is an element of the class U (2) because the sequence {ak log α k} is uniformly distributed modulo 1 [3, Exercise 3.14].
On the other hand, this sequence does not belong to the set of sequences of [15].
exists for all but at most countably many ε > 0.

Auxiliary results
For the proof of universality for the function ζ(s, F ), we will use the probabilistic approach.Denote by B(X) the Borel σ-field of the space X.Let P n , n ∈ N, and P be the probability measures on (X, B(X)).We remind that P n , as n → ∞, converges weakly to P if, for every real continuous bounded function g on X, Denote by H(D F ) the space of analytic functions on D F endowed with the topology of uniform convergence on compacta.The proof of universality theorems is based on the weak convergence for as N → ∞.For the statement of a limit theorem for P N,F , we need some notation.Let P be the set of all prime numbers, and let γ denote the unit circle on the complex plane.Define the set where γ p = γ for all p ∈ P. With the product topology and pointwise multiplication, the infinite-dimensional torus Ω is a compact topological Abelian group, therefore, on (Ω, B(Ω)), the probability Haar measure m H can be defined.This gives the probability space (Ω, B(Ω), m H ). Denote by ω(p) the projection of an element ω ∈ Ω to the coordinate space γ p , p ∈ P, and, on the probability space (Ω, B(Ω), m H ), define the .
Let P ζ,F stand for the distribution of ζ(s, ω, F ), i.e., Now we state the main result of this section.
https://www.mii.vu.lt/NATheorem 5. Suppose that ϕ ∈ U (k 0 ).Then P N,F converges weakly to P ζ,F as N → ∞.Moreover, the support of P ζ,F is the set We divide the proof of Theorem 5 into several lemmas.We start with the Weyl criterion.
Proof of the lemma can be found, for example, in [3].
Lemma 2. Suppose that ϕ ∈ U (k 0 ).Then Q N converges weakly to the Haar measure m H as N → ∞.
Proof.We apply the Fourier transform method.It is well known that the dual group of Ω is isomorphic to the group where Z p = Z for all p ∈ P.An element k = {k p : k p ∈ Z, p ∈ P} of D, where only a finite number of integers k p are distinct from zero, acts on Ω by where the sign " " means that only a finite number of integers k p are distinct from zero.Hence, the characters are of the form p∈P ω kp (p), therefore, the Fourier transform g N (k) of Q N is given by the formula Thus, by the definition of Q N , Obviously, Since the set {log p: p ∈ P} is linearly independent over the field of rational numbers Q, we have that p∈P k p log p = 0 for k = 0. Therefore, since ϕ ∈ U (k 0 ), in the case k = 0, the sequence ϕ(k) 2π p∈P k p log p: k k 0 is uniformly distributed modulo 1.Thus, by Lemma 1 with m = −1 and (1), we find that, for k = 0, lim This and (2) show that g N (k), as N → ∞, converges to the Fourier transform of the Haar measure m H , and the lemma is a consequence of a continuity theorem for probability measures on compact groups.
Lemma 2 implies a limit theorem in the space of analytic functions for a certain absolutely convergent Dirichlet series.This theorem is very important for proving Theorem 5, therefore, we give its precise statement.
We extend the functions ω(p), p ∈ P, to the set N by Let θ > 1/2 be a fixed number.For m, n ∈ N, define the series Then, the latter series are absolutely convergent for σ > κ/2.Let the function u n,F : Ω → H(D F ) be given by the formula u n,F (ω) = ζ n (s, ω, F ). Since the series for ζ n (s, ω, F ) is absolutely convergent for σ > κ/2, the function u n,F is continuous, thus, it is (B(Ω), B(H(D F )))-measurable.Hence, P n,F = m H u −1 n,F , where https://www.mii.vu.lt/NA The above remarks, Lemma 2, and Theorem 5.1 of [1] lead to Lemma 3. Suppose that ϕ ∈ U (k 0 ).Then P N,n,F converges weakly to P n,F as N → ∞.
Our next aim is to prove that P N,F , as N → ∞, converges weakly to the limit measure P F of P n,F as n → ∞.For this, we need some mean square results for the function ζ(s, F ). Lemma 4. Suppose that ϕ ∈ U (k 0 ), and σ, κ/2 < σ < (κ + 1)/2, is fixed.Then, for all τ ∈ R, Proof.It is well known that, for fixed σ, κ/2 < σ < (κ + 1)/2, Let X > 1.Since the function ϕ(t) is increasing and continuously differentiable, we have that By estimate (3), Since ϕ ∈ U (k 0 ), the latter estimate together with (4) shows that Lemma 4 together with Gallagher's lemma, which connects the continuous and discrete mean squares of some functions, allows to estimate the discrete mean square For convenience, we state Gallagher's lemma, see [14,Lemma 1.4].
Lemma 5. Suppose that T 0 , T δ > 0 are real numbers, and Let S(x) be a complex-valued continuous function on [T 0 , T + T 0 ] having a continuous derivative on (T 0 , T + T 0 ).
. Lemma 6. Suppose that ϕ ∈ U (k 0 ), and σ, κ/2 < σ < (κ + 1)/2, is fixed.Then, for t ∈ R, Proof.An application of the Cauchy integral formula and Lemma 4 gives, for κ/2 < σ < (κ + 1)/2, the bound Actually, in view of the Cauchy integral formula, where L is the circle with a center σ lying in D. Then https://www.mii.vu.lt/NAHence, in view of Lemma 4, We apply Lemma 5 with and δ = 1.Then, clearly, N δ (x) = 1, and, in view of Lemma 5 with S(τ ) = ζ(σ + it + iϕ(τ ), F ), we have . This, Lemma 4, and estimate (5) prove the lemma.Now we are ready to approximate ζ(s, F ) by ζ n (s, F ) in the mean.For where {K l : l ∈ N} ⊂ D F is a sequence of compact subsets such that Then ρ is the metric in H(D F ) inducing its topology of uniform convergence on compacta.
Lemma 7. Suppose that ϕ ∈ U (k 0 ).Then Let K be an arbitrary compact subset of D. Then, using the above integral representation and the residue theorem, we find that where σ < 0, κ/2 < σ < (κ + 1)/2, and t is bounded by a constant depending on K. Now an application of Lemma 6 and ( 6) implies the equality This and the definition of the metric ρ prove the lemma.
Proof of Theorem 5. Let θ N be a random variable defined on a certain probability space with the measure µ and having the distribution Consider the H(D F )-valued random element We recall that P n,F is the limit measure in Lemma 3.Then, in view of Lemma 3, where → means the convergence in distribution, and X n,F is the H(D F )-valued random element with distribution P n,F .Using the absolute convergence of the series for ζ n (s, F ) and ( 7), we prove by using the method of [4] that the family of probability measures https://www.mii.vu.lt/NA { P n,F : n ∈ N} is tight.Hence, by Theorem 6.1 of [1], it is relatively compact.Therefore, each subsequence of { P n,F } contains a subsequence { P nr,F }, which converges weakly to a certain probability measure On the probability space of the random variable θ N , define the Then the application of Lemma 7 shows that, for every ε > 0, From this, ( 7), (8), and Theorem 4.2 of [1] it follows that This means that P N,F converges weakly to P F as N → ∞.On the other hand, (9) shows that the measure P F is independent of the sequence { P nr,F }.Since the family { P n,F } is relatively compact, hence we have, by Theorem 2.3 of [1], that or equivalently, P n,F converges weakly to P F as n → ∞.It remains to identity the measure P F .For this, usually, elements of the ergodic theory are applied.However, we use a very simple observation.It is known [7,17] that as T → ∞, converges weakly to the limit measure P F of P n,F and that P F = P ζ,F .Moreover, the support of P ζ,F is the set S F .Therefore, P N,F also converges weakly to P ζ,F as N → ∞.This, the definitions of P N,F and G ε , and (10) show that lim inf

Proofs of universality theorems
By the Mergelyan theorem on the approximation of analytic functions by polynomials [13], we can choose the polynomial p(s) to satisfy the inequality This inequality together with (11) proves Theorem 3. Hence, ∂ G ε1 ∩ ∂ G ε2 = ∅ for ε 1 = ε 2 .Therefore, the set G ε is a continuity set of the measure P ζ,F for all but at most countably many ε > 0. Using Theorem 5 and the equivalent of weak convergence of probability measures in terms of continuity sets [1, Thm.2.1], we obtain that lim for all but at most countably many ε > 0. In view of (12), if g ∈ G ε , then g ∈ G ε .Thus, G ε ⊂ G ε .Therefore, in virtue of (10), P ζ,F ( G ε ) > 0. Combining this with (13) and the definitions of P N,F and G ε proves Theorem 4. https://www.mii.vu.lt/NA b, c, d ∈ Z, ad − bc = 1 be the full modular group.The function F (z) is called a holomorphic cusp form of weight κ for SL(2, Z) if F (z) is holomorphic for Im z > 0, for all a b c d ∈ SL(2, Z), satisfies the functional equation F az + b cz + d = (cz + d) κ F (z), and, at infinity, has the Fourier series expansion F (z) = ∞ m=1 c(m)e 2πimz .

Proof of Theorem 4 .
Define the setG ε = g ∈ H(D): sup s∈K g(s) − f (s) < ε .Then we have that the boundary∂ G ε of G ε is the set g ∈ H(D): sup s∈K g(s) − f (s) = ε .