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 Lp integrability properties of the modified zeta functions on the abscissa for p[1,∞].