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

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发表日期：

2026年5月19日

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Jongho Yang

摘要

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 $L^2$ $L^{2}$ . Later, Balazard and Saias posed a question regarding the coefficients of these natural Beurling functions, which Weingartner answered using the sequence space $l^2$ $l^{2}$ . In this paper, we extend Weingartner’s theorem to the Banach space $L^p$ $L^{p}$ , a setting that is more difficult to handle than the Hilbert space $l^2$ $l^{2}$ . We also show that the Beurling functions form the unique best minimal system for approximating the characteristic function in $L^p$ $L^{p}$ .