Modified zeta functions as kernels of integral operators |
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Authors: | Jan-Fredrik Olsen |
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Institution: | Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway |
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Abstract: | The modified zeta functions ∑n∈Kn−s, where K⊂N, converge absolutely for . These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of C 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 I⊂R. We also consider the special case when the set K⊂N is assumed to have arithmetic structure. In particular, we look at local Lp integrability properties of the modified zeta functions on the abscissa for p∈1,∞]. |
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Keywords: | Zeta function Integral operator Frame theory Tauberian theory |
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