Joint value-distribution of shifts of the Riemann zeta-function

We prove that any non-zero complex values $z_1,\ldots,z_n$ can be approximated by the following integral shifts of the Riemann zeta-function $\zeta(s+id_1\tau),\ldots,\zeta(s+id_n\tau)$ for infinitely many $\tau$, provided $d_1,\ldots,d_n\in\mathbb{N}$ and $s$ is a fixed complex number lying in the right open half of the critical strip.


Introduction
In the 80's of the last century Bagchi (see [1,2]) observed that the classical Riemann Hypothesis is equivalent to the fact that for every ε > 0 and every compact set K ⊂ D := {s ∈ C : 1/2 < Re(s) < 1} with connected complement we have (1) lim inf where meas denotes the Lebesgue measure on R. In other words (1) means that the set of real τ satisfying max s∈K |ζ(s + iτ ) − ζ(s)| < ε has a positive lower density. Moreover, let us note that in the language of topological dynamics Inequality (1) says that the Riemann Hypothesis is equivalent to the strong recurrence of the Riemann zeta-function (see [8]). One implication in Bagchi's observation is an immediate consequence of the so-called universality theorem due to Voronin [17], which is a generalization of the work of Bohr and his collaborators (see [3,4,5]) on denseness theorems in C of values of the Riemann zeta-function and states that for any non-vanishing and continuous function f (s) on K, analytic in the interior of K, and every ε > 0, we have The converse implication is a combination of Rouché's theorem and the well-known zero density theorem for the Riemann zeta-function. An interesting generalization of (1) was suggested by Nakamura in [10], where he introduced the problem of finding all d 1 , d 2 , . . . , d n such that for every compact set K ⊂ D with connected complement and every ε > 0 we have (2) lim inf Notice that (2) holds for n = 2 and d 1 = 0, d 2 = 0 if and only if the Riemann Hypothesis is true. Interestingly it turns out (see [10,11,12,13]) that for n = 2 the case d 1 = 0, d 2 = 0 is the most The author was partially supported by the grant no. 2016/23/D/ST1/01149 from the National Science Centre.
1 difficult one as it was shown that (2) holds for n = 2 and every non-zero real d 1 , d 2 . A common strategy of showing (2) is to prove a joint version of universality for ζ(s + id 1 τ ), . . . , ζ(s + id n τ ), namely to prove that for any non-vanishing and continuous functions f 1 (s), . . . , f n (s) on K, analytic in the interior of K, and every ε > 0 we have Obviously, the truth of (3) implies (2). The problem of finding all parameters d 1 , d 2 satisfying (3) for n = 2 was also completely solved in the aforementioned series of papers by Nakamura and the author, where it was proved that (3) holds for n = 2 and any real d 1 , d 2 with 0 = d 1 , d 2 and d 1 = ±d 2 . Note that it is indeed a complete answer for n = 2 as one cannot expect that (3) holds for d 1 = ±d 2 . Unfortunately, the situation for n ≥ 3 is much more complicated and so far we know only (see [10]) that (3) holds if d 1 = 1, d 2 , . . . , d n are algebraic real numbers linearly independent over Q, which is an easier case since for such d k 's, in view of Baker's theorem in transcendental number theory, the set I(d 1 , . . . , d n ) := {d k log p : p is prime, 1 ≤ k ≤ n} is linearly independent over Q and one can follow straightforwardly the proof of Voronin's theorem in order to get (3). However, if the set I(d 1 , . . . , d n ) is not linearly independent over Q, we need a new approach or at least a significant modification of known proofs of universality, since linear independence of the set I(d 1 , . . . , d n ) is crucial in several steps of these proofs.
In the paper we focus on considering the following weaker version of (3) in the flavor of denseness theorems due to Bohr, where K is a singleton and d 1 , . . . , d n are positive integers. Note that in this case the set I(d 1 , . . . , d n ) is far from being linearly independent over the field of rational numbers. Theorem 1.1. Let d 1 < d 2 < · · · < d n ∈ N and s = σ + it with σ ∈ (1/2, 1] and t > 0. Then, for every z 1 , z 2 , . . . , z n ∈ C and every ε > 0 the set of all positive real τ satisfying max 1≤k≤n | log ζ(s + id k τ ) − z k | < ε has a positive lower density.
Obviously, the theorem easily implies that (3) holds for K = {s} and f k (s) = z k = 0. Moreover, it is worth mentioning that the proof of Theorem 1.1 presented in the paper might be simplified in several places under the assumption that d 1 , . . . , d n are integers. However we decided to make all steps in the proof as general as possible since it might be interesting for the reader to see that in fact the proof of the above theorem is valid for arbitrary positive real d 1 , . . . , d n except only one step where we use the implication which is trivially true for d k ∈ Z, but it does not necessary hold if d k / ∈ Z.

Denseness lemma
In this section we prove the following denseness lemma, which holds for all positive real d 1 , . . . , d n .
The proof of the lemma essentially based on the following Pechersky's [14] generalization of the Riemann rearrangement lemma to Hilbert spaces. 14]). Let H be a real Hilbert space with the inner product (·|·) and a sequence (u n ) n∈N ⊂ H be such that ∞ n=1 u n 2 H < ∞. Moreover, suppose that for every e ∈ H with e = 1 the series ∞ n=1 u n , e are conditionally convergent after suitable permutation of terms. Then, for every v ∈ H there exists a permutation (n k ) such that ∞ k=1 u n k = v. Moreover, we use the following effective version of Kronecker approximation theorem due to Chen. Henceforth, · will denote the distance to the nearest integer. Lemma 2.3 (Chen, [6]). Let λ 1 , . . . , λ n and α 1 , . . . , α n be real numbers and assume the following property: for all integers u 1 , . . . , u n with |u j | ≤ M , the assertion u 1 λ 1 + · · · + u n λ n ∈ Z implies that u 1 α 1 + · · · + u n α n is an integer. Then for all positive real numbers δ 1 , . . . , δ n and for any integers Proof. Applying Chen's lemma for Obviously, we have sin 2 π 2(M +1) ≤ 2 ω . Moreover, since M depends only on ω and Λ depends only on M and d 1 , . . . , d n , there is a sufficiently large T such that M n 2ΛT ≤ 2 ω , which completes the proof.
The need for the application of Lemma 2.2 forces us to consider the distribution of zeros of functions being linear combinations of exponential functions, which is well understood by the following result due to Wilder.
Proof. For the proof we refer to [7, Theorem 1].
Proof of Lemma 2.1. Let θ pm = m L , where p m denotes the m-th prime number and L is an arbitrary integer satisfying L > n − 1 + d n − d 1 . We shall use Lemma 2.2 for the real Hilbert space H = C n equipped with the standard inner product and the sequence Obviously, ∞ m=1 u m 2 < ∞, since σ > 1/2. Thus, in order to apply Lemma 2.2, it suffices to show that for every e = (a 1 , . . . , a n ) ∈ C n with e H = 1 there are two permutations of the series p (u m |e) tending to +∞ and −∞, respectively. In fact, we show only the existence of a permutation of the series, which diverges to +∞, since the case −∞ is similar and can be left to the reader.
Let us note that and consider the function f (x) = n k=1 a k e(d k x), x ∈ R. Putting A = n k=1 |a k | 2 , one can easily deduce from the Cauchy-Schwarz inequality that for every x ∈ R we have . Now, from Corollary 2.4, we see that there is a positive integer N such that every interval of length N contains an integer h satisfying n k=1 hd k 2 ≤ ε 2 (2π) 2 A . Therefore, dividing the interval (0, ∞) into disjoint intervals I l , l ∈ N, of length N , one can find integers τ l ∈ I l satisfying n k=1 τ l d k 2 ≤ ε 2 (2π) 2 A . Hence, for every x ∈ R and l ∈ N, we have Now, let us observe that the function f (x) has at most n − 1 + d n − d 1 zeros in the interval [0, 1]. Hence, by the choice of L, there is m 0 such that 0 ≤ m 0 ≤ L − 1 and f ( m 0 L ) = c 0 = 0. Thus for every τ l we have so each interval [m 0 + kLN, m 0 + (k + 1)N L], k = 1, 2, . . ., contains an integer m ′ k satisfying f ( m ′ k L ) = c 0 + ξ l with |ξ l | ≤ ε. Assume that c 0 = |c 0 |e iϕ . Then for every m ′ k and m ∈ N such that 2πm Hence it remains to notice that simple calculations and the prime number theorem imply that so using the fact that the harmonic series is divergent completes the proof.

Approximation by a finite sum over primes
In this section we use the following Tsang's lemma in order to show that the logarithm of the Riemann zeta-function can be approximated by a truncated Dirichlet series over primes for sufficiently many arguments from any vertical line lying in the right open half of the critical strip.
Then, for sufficiently large T , we have Proof. First let us observe that using Tsang's lemma and a suitable version of Hilbert's inequality due to Montgomery and Vaughan (see [9]) give for some c ′ > 0 and k = 1, 2, . . . , n. Hence for every ε 0 > 0 and sufficiently large T we have

Proof of the main theorem
In order to prove our main theorem we need the following crucial lemma which allows us to connect Lemma 2.1 and Lemma 3.2.
Case 2: p = q, 0 = d k ∈ Q. Similarly to the Case 1, by the unique factorization of integers, Using the Six Exponentials Theorem one can prove (see [  Now, let us note that B = p/q for some p, q / ∈ M , so p, q > y. Hence, taking sufficiently large y and using the above lemma we obtain that (5) never holds, which means that |x − y| > 0 in Case 3 as well.
Therefore, Now, we are in the position to complete the proof of our theorem. Let us emphasize that the choice of such d is possible since d 1 , . . . , d n ∈ N, and it is the only step in the proof where we need to assume that d 1 , . . . , d n are integers. Let us notice that for τ ∈ A d (T ) we have max 1≤k≤n p∈M 1 p s+id k τ − z k < 2ε.