Abstract
We prove that any complex values z_1,ldots ,z_n can be approximated by the following integral shifts of the Riemann zeta-function zeta (s+id_1tau ),ldots ,zeta (s+id_ntau ) for infinitely many tau , provided d_1,ldots ,d_nin {mathbb {Q}} are distinct and positive, and s is a fixed complex number lying in the right open half of the critical strip.
Highlights
In the 80’s of the last century Bagchi 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 lim inf 1 meas τ ∈ [0, T ] : max |ζ(s + iτ ) − ζ(s)| < ε > 0, (1)
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
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 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 lim inf 1 meas τ ∈ [0, T ] : max |ζ(s + iτ ) − f (s)| < ε > 0
Summary
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 lim inf 1 meas τ ∈ [0, T ] : max |ζ(s + iτ ) − ζ(s)| < ε > 0, (1). T →∞ T s∈K where meas denotes the Lebesgue measure on R. In other words (1) means that the set of real τ satisfying maxs∈K |ζ(s + iτ ) − ζ(s)| < ε has a positive lower density. 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 [9]).
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