Modified zeta functions as kernels of integral operators

Abstract

The modified zeta functions ∑ n ∈ K n − s , where K ⊂ N , converge absolutely for Re s > 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 L 2 ( 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 L p integrability properties of the modified zeta functions on the abscissa Re s = 1 for p ∈ [ 1 , ∞ ] .