Modified zeta functions as kernels of integral operators

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,∞].