Modified zeta functions as kernels of integral operators

Abstract

The modified zeta functions ${\sum }_{n\in K}{n}^{-s}$, where $K\subset \mathbb{N}$, 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 $I\subset \mathbb{R}$. We also consider the special case when the set $K\subset \mathbb{N}$ 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 $p\in [1,\infty ]$.