A generalization of Weingartner’s theorem related to the Riemann hypothesis

Abstract

In the study of the Riemann hypothesis, Báez-Duarte provided a strong version of the Nyman–Beurling criterion by considering the approximation of the characteristic function using natural Beurling functions in \(L^2\). Later, Balazard and Saias posed a question regarding the coefficients of these natural Beurling functions, which Weingartner answered using the sequence space \(l^2\). In this paper, we extend Weingartner’s theorem to the Banach space \(L^p\), a setting that is more difficult to handle than the Hilbert space \(l^2\). We also show that the Beurling functions form the unique best minimal system for approximating the characteristic function in \(L^p\).