Joint universality and generalized strong recurrence with rational parameter

摘要

We prove that, for every rational d not equal 0, +/- 1 and every compact set K subset of {s is an element of C : 1/2 < Re(s) < 1} with connected complement, any analytic non-vanishing functions f(1), f(2) on K can be approximated, uniformly on K, by the shifts zeta(s + i tau) and zeta(s + id tau), respectively. As a consequence we deduce that the set of tau satisfying vertical bar zeta(s + i tau) - zeta (s + id tau)vertical bar < epsilon uniformly on K has a positive lower density for every d not equal 0. (c) 2016 Elsevier Inc. All rights reserved.