Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros

: Let 0 < γ 1 < γ 2 < · · · (cid:54) γ k (cid:54) · · · be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ ( s ) . Using a certain estimate on the pair correlation of the sequence { γ k } in the intervals [ N , N + M ] with N 1/2 + ε (cid:54) M (cid:54) N , we prove that the set of shifts ζ ( s + ih γ k ) , h > 0, approximating any non-vanishing analytic function deﬁned in the strip { s ∈ C : 1/2 < Re s < 1 } with accuracy ε > 0 has a positive lower density in [ N , N + M ] as N → ∞ . Moreover, this set has a positive density for all but at most countably ε > 0. The above approximation property remains valid for certain compositions F ( ζ ( s )) .


Introduction
The Riemann zeta-function ζ(s), s = σ + it, is defined, for σ > 1, by where the infinite product is taken over all prime numbers, and has analytic continuation over the whole complex plane, except for the point s = 1 which is a simple pole with residue 1. The function ζ(s) and its value distribution play an important role not only in analytic number theory but in mathematics in general. It is well known by a Bohr and Courant work [1] that the set of values of ζ(σ + it) with any fixed σ ∈ (1/2, 1] is dense in C. Voronin obtained [2] the infinite-dimensional version of the Bohr-Courant theorem, proving the so-called universality of ζ(s). This means that every non-vanishing analytic function in the strip D = {s ∈ C : 1/2 < σ < 1} can be approximated by shifts ζ(s + iτ). We recall the modern version of the Voronin theorem. 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 non-vanishing functions on K that are analytic in the interior of K. Then, for K ∈ K, f (s) ∈ H 0 (K) and every ε > 0, the inequality lim inf T→∞
The above theorem is of continuous type because τ in shifts ζ(s + iτ) can take arbitrary real values. If τ runs over a certain discrete set, then we have the discrete universality that was proposed in [7]. Denote by #A the cardinality of a set A, and suppose that N runs over the set of non-negative integers. If K and f (s) are as above, then we have, for h > 0 and ε > 0, Approximations of analytic functions by more general discrete shifts were considered in [8][9][10]. Denote by γ 1 < γ 2 < · · · γ k · · · the positive imaginary parts of non-trivial zeros ρ k = β k + iγ k of the function ζ(s). Discrete universality theorems with shifts ζ(s + ihγ k ) were obtained in [11,12]. In [11], for this the Riemann hypothesis was used, while in [12], the weak form of the Montgomery pair correlation conjecture [13] was involved. More precisely, the estimate, for c > 0, was required. Analogical results for more general functions were given in [14,15]. On the other hand, all above theorems are non-effective in the sense that any concrete shift approximating a given analytic function is not known. This shortcoming leads to the idea of universality in intervals as short as possible containing τ with approximating property. The first result in this direction was obtained in [16].
The aim of this paper is the universality of the function ζ(s) in short intervals with shifts ζ(s + ihγ k ). In this case, the estimate (1) is not sufficient. Therefore, for N 1/2+ε M N with ε > 0, we use the following hypothesis: which, as estimate (1), also gives a certain information on the pair correlation of non-trivial zeros, differently from estimate (1), however, in short intervals.

Theorem 2.
Suppose that N 1/2+ε M N, and estimate (2) are true. Let K ∈ K and f (s) ∈ H 0 (K). Then, for every ε > 0 and h > 0, Moreover, "lim inf" can be replaced by "lim" for all but at most countably many ε > 0. is a continuous operator such that F(S) ⊃ H a 1 ,...,a r ;F (D). For r = 1, let K ∈ K and f (s) be a continuous = a 1 function on K, and analytic in the interior of K. For r 2, let K be an arbitrary compact subset of D, and f (s) ∈ H a 1 ,...,a r ;F (D). Then, for every ε > 0 and h > 0, Moreover "lim inf" can be replaced by "lim" for all but at most countably many ε > 0.
For example, the operators F(g) = sin g and F(g) = sinh g satisfy the hypotheses of Theorem 3 with a 1 = −1 and a 2 = 1.
The proofs of Theorems 2 and 3 use probabilistic limit theorems for measures in the space H(D). Denote by B(X) the Borel σ-field of the space X. The main limit theorem will be proved for as N → ∞. We divide its proof into four sections.

A Limit Theorem on the Torus
Denote by γ the unit circle on the complex plane, by P the set of all prime numbers, and define the set where γ p = γ for all p ∈ P. With the product topology and pointwise multiplication, the torus Ω is a compact topological Abelian group. Therefore, on (Ω, B(Ω)), the probability Haar measure m H can be defined, and we have the probability space (Ω, B(Ω), m H ). Denote by ω(p) the pth component of an element ω ∈ Ω, p ∈ P.
In this section, we will prove a limit theorem for Before the statement of a limit theorem for Q N,M,h as N → ∞, we will recall some useful results that will be used in its proof. Denote by N(T) the number of non-trivial zeros of ζ(s) in the region {s ∈ C : 0 < t < T}.
An application of Lemma 1 gives Therefore, Hence, This together with Equation (3)  Proof. Denote by g N,M,h (k), k = (k p : k p ∈ Z, p ∈ P), the Fourier transform of Q N,M,h , i.e., where the star " * " means that only a finite number of integers k p are distinct from zero. Thus, by the definition of Q N,M,h , Clearly, Now, suppose that k = 0. Since the set {log p : p ∈ P} is linearly independent within the field of rational numbers Q, in that case we have Thus, we will estimate the sum It is easily seen that Therefore, by Lemma 2 and estimate (2), This, and estimates (7) and (8) show that Lemma 4 with x = exp{ha} implies Therefore, in view of estimate (9), Thus, by Equation (5), This together with Equation (6) shows that and the lemma is proved because the right-hand side of the latter equality is the Fourier transform of the measure m H .

A Limit Theorem for Absolutely Convergent Series
Let θ > 1/2 be a fixed number, and v n (m) = exp{−(m/n) θ } for m, n ∈ N. Extend the function ω(p) to the set N by setting Then the latter series are absolutely convergent for σ > 1/2 [5]. Consider the function u n : Ω → H(D) defined by u n (ω) = ζ n (s, ω).
The absolute convergence of the series ζ n (s, ω) implies the continuity of u n . For A ∈ B(H(D)), define Then ζ(s, ω) is an H(D)-valued random element on the probability space (Ω, B(Ω), m H ) [5]. We recall that the latter infinite product, for almost all ω, is uniformly convergent on compact subsets K ⊂ D. Denote by P ζ the distribution of the random element ζ(s, ω), i.e., (H(D)).
The following statement is very important. Proposition 1. The probability measure V n converges weakly to measure P ζ as n → ∞.

Proof. For
It is known that R T , as T → ∞, converges weakly to P ζ [5]. Moreover, R T , as T → ∞, and V n , as n → ∞, converge weakly to the same probability measure on (H(D), B(H(D))). Thus, V n converges weakly to P ζ as n → ∞.

Mean Square Estimates in Short Intervals
To derive the weak convergence of P N,M,h from that of P N,M,n,h as N → ∞, the estimate for We will use the following mean square estimate in short intervals.
Lemma 6. Suppose that N 1/2+ε M N and estimate (2) is true. Then, for every fixed σ, 1/2 < σ < 1, h > 0 and t ∈ R, Proof. We will apply the Gallagher lemma connecting discrete mean squares with those continuous of some functions; for the proof, see Lemma 1.4 of [22]. Let T 0 , T δ > 0 be real numbers, T = ∅ be a finite set in the interval 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 ). Then the Gallagher lemma asserts that We apply the Gallagher lemma for the function ζ(s + ikhγ k + it). In our case δ = c/ log N, T 0 = hγ N − δ/2, T = hγ N+M − hγ N + δ/2 and T = {hγ N , hγ N+1 , . . . , hγ N+M }. By estimate (2), we have Now, an application of the Gallagher lemma gives The estimate (4) gives with certain c h > 0 If c h (M/ log M) + |t| hγ N , then, in view of Lemma 5, the right-hand side of (13) is If c h (M/ log M) + |t| > hγ N , then Thus, in this case, This together with estimate (13) shows that Estimate (14) and an application of the Cauchy integral formula lead to the bound This, estimate (14) and (12) prove the lemma. Now, we are ready to state an approximation lemma.

Approximation in the Mean
Denote by ρ the metric in H(D) which induces the topology of uniform convergence on compacta. More precisely, for g 1 , g 2 ∈ H(D), where {K l : l ∈ N} ⊂ D is a sequence of compact subsets such that Proof. In view of the definition of the metric ρ, it suffices to show that, for every compact K ⊂ D, Thus, let K ⊂ D be a fixed compact set. Denote the points of K by s = σ + iv, and fix ε > 0 such that 1/2 + 2ε σ 1 − ε for s ∈ K. It is known [5] that where l n (s) = s θ Γ(s/θ)n s , Γ(s) is the Euler gamma-function, and θ comes from the definition of v n (m). Let θ 1 > 0. From this, we have Therefore, as in the proof of Lemma 12 of [16], we find that Denote by c 1 , c 2 , . . . positive constants. In view of the well-known estimate we find that |l n (1/2 + ε − s + it)| |1/2 + ε − s + it| n −ε exp{−c 2 |t − v|} K,ε n −ε exp{−c 3 |t|}.

A Limit Theorem for ζ(s)
Using the results of Sections 3 and 4 leads to a limit theorem for P N,M,h . Moreover, let X n = X n (s) be the H(D)-valued random element with the distribution V n . Then, by Theorem 5, where D − → denotes the convergence in distribution. Moreover, by Proposition 1, Define one more H(D)-valued random element Then, using Lemma 7, we find that, for every ε > 0, ρ(ζ(s + ihγ k ), ζ n (s + ihγ k )) = 0. Now, this, Equations (19) and (20) together with Theorem 4.2 of [20] show that and theorem is proved.
For A ∈ B(H(D)), define

Proof of Universality
Theorems 2 and 3 are derived from Theorem 6 and Corollary 1, respectively, by using the Mergelyan theorem on the approximation of analytic functions by polynomials [23].
Proof of Theorem 2. We recall that S = {g ∈ H(D) : either g(s) = 0 for all s ∈ D, or g(s) ≡ 0} , It is well known, see, for example, [5], that the support of the measure P ζ is the set S. Define the set where p(s) is a polynomial. Obviously, e p(s) ∈ S. Therefore, G ε is an open neighbourhood of an element of the support of the measure P ζ . Thus, by a property of the support, This, Theorem 6 and the equivalent of weak convergence in terms of open sets show that Hence, by the definition of P N,M,h and G ε , Now, we apply the Mergelyan theorem and choose the polynomial p(s) satisfying This and inequality (22) prove the first part of the theorem.
To prove the second part of the theorem, define the set Then the setĜ ε is a continuity set of the measure P ζ for all but at most countably many ε > 0. This remark, Theorem 6 and the equivalent of weak convergence of probability measures in terms of open sets show that lim for all but at most countably many ε > 0. Inequality (23) implies the inclusion G ε ⊂Ĝ ε . Therefore, in view of inequality (21), we have P ζ (Ĝ ε ) > 0. This, Equation (24) and the definitions of P N,M,h and G ε prove the second part of the theorem.
Proof of Theorem 3. Denote by S F the support of the measure P ζ F −1 . We observe that S F contains the closure of the set H a 1 ,...,a r ;F (D). Actually, let g ∈ H a 1 ,...,a r ;F (D) and G be any open neighborhood of g. Then the set F −1 G is open as well, and lies in S. Hence, P ζ (F −1 G) > 0 because S is the support of P ζ . Therefore, This shows that S F contains the set H a 1 ,...,a r ;F (D) and its closure. Case r = 1. By the Mergelyan theorem, there exists a polynomial p(s) such that Then, p(s) = a 1 for all s ∈ K if ε is small enough. Therefore, by the Mergelyan theorem again, we find a polynomial q(s) such that is an open subset of S F . Hence, This inequality together with Corollary 1, inequalities (25) and (26) prove the theorem in the case of the lower density.
In the case of density, consider the setĜ ε defined in the proof of Theorem 2 which is a continuity set of the measure P ζ F −1 for all but at most countably many ε > 0. Therefore, by Corollary 1, lim N→∞ P N,M,h,F (Ĝ ε ) = P ζ F −1 (Ĝ ε ).
Case r 2. In this case, the function f (s) lies in S F . Therefore, the Mergelyan theorem is not needed, and the theorem follows immediately from Corollary 1.

Conflicts of Interest:
The authors declare no conflict of interest.