Non-universality of the Riemann zeta function and its derivatives when σ ≥ 1

Let zeta (s) be the Riemann zeta function. In 1911, Bohr showed that the set {zeta (sigma + i tau) : sigma > 1, tau is an element of R} is dense in C. By Voronin's denseness theorems in 1972, the sets {(zeta(sigma + i lambda(1) +i tau), ..., + zeta(sigma + i lambda(n) +i tau)) sigma >= 1,tau is an element of R} with distinct lambda(1), ..., lambda(n) is an element of R and {(zeta(sigma + i tau),zeta'(sigma +i tau), ..., zeta((n-1))(sigma + i tau)) : sigma >= 1, tau is an element of R} are dense in C-n. By Voronin's universality theorem, for any fixed 1/2 < sigma < 1 and any non-negative integer k, the set {zeta((k))(sigma,tau): tau is an element of R} is dense in C[a, b], where zeta((k))(sigma,tau) (t) := zeta((k))(sigma+ it + i tau), t is an element of [a, b]. In the present paper, we prove that the set {zeta((k))(sigma,tau) : sigma >= 1, tau is an element of R} boolean AND C[a, b] is not dense in C [a, b]. (C) 2019 Elsevier Inc. All rights reserved.