Modified zeta functions as kernels of integral operators

Abstract

The modified zeta functions $\sum_{n \in K} n^{-s}$, where $K \subset \N$, converge absolutely for $\Re s > 1/2$. 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 $I \subset \R$. We also consider the special case when the set $K \subset \N$ 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 $p \in [1,\infty]$.