A generalization of Beurling's criterion for the Riemann hypothesis

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• 作者：Jongho Yang ^{cachya@korea.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11Mxx
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：164
• 期：Complete
• 页码：299-302
• 全文大小：202 K

文摘

It is known that the Riemann hypothesis holds if and only if the function χ_(0,1)${\chi }_{(0,1)}$ can be approximated by linear combinations of u_α$u_{\alpha }$ in L²(0,1)$L^2(0,1)$. Here u_α(x)$u_{\alpha }(x)$ is defined by [α/x]−α[1/x]$[\alpha /x]-\alpha [1/x]$ for 0<α<1$0<\alpha <1$. In this note we generalize the Beurling's equivalent condition by replacing the function χ_(0,1)${\chi }_{(0,1)}$ with χ_(a,b)${\chi }_{(a,b)}$ for any 0≤a<b≤1$0\le a<b\le 1$.