Modified zeta functions as kernels of integral operators

The modified zeta functions n∈K n−s, where K ⊂ N, converge absolutely for Res > 1. 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 Re s = 1 for p ∈ [1,∞]. © 2010 Elsevier Inc. All rights reserved