Modified zeta functions as kernels of integral operators

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• 作者：Jan-Fredrik Olsen
• 关键词：Zeta function ; Integral operator ; Frame theory ; Tauberian theory
• 刊名：Journal of Functional Analysis
• 出版年：2010
• 出版时间：15 July 2010
• 年：2010
• 卷：259
• 期：2
• 页码：359-383
• 全文大小：235 K

文摘

The modified zeta functions ∑_nKn^−s, where , converge absolutely for . These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of with a single pole at s=1. Our main result is a characterisation of the modified zeta functions that have pole-like behaviour at this point. This behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces L²(I) for symmetric and bounded intervals . We also consider the special case when the set is assumed to have arithmetic structure. In particular, we look at local L^p integrability properties of the modified zeta functions on the abscissa for p[1,∞].