Geometric sequences and zero-free region of the zeta function

Abstract

Let N be the linear space of functions Sigma(n)(k=1) a(k)rho(theta(k)/x) with a condition Sigma(n)(k=1) a(k)theta(k) = 0 for 0 < theta(k) <= 1. Here rho(x) denotes the fractional part of x. Beurling pointed out that the problem of how well a constant function can be approximated by functions in N is closely related to the zero-free region of the Riemann zeta function. More precisely, Baez-Duarte gave a zero-free region related to a L-p-norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the L-infinity-norm estimation of a constant function and give a wider zero-free region than that of the Baez-Duarte result. (c) 2018 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.