Zeta-Functions for Non-Minimal Operators

TL;DR

This paper computes the zeta-function at zero for invariant non-minimal vector and tensor operators on symmetric spaces, providing explicit formulas and asymptotic expansions for various dimensions.

Contribution

It introduces a method to evaluate $\

Findings

Explicit $\

paper_type

empirical

Abstract

We evaluate zeta-functions $\zeta(s)$ at $s=0$ for invariant non-minimal 2nd-order vector and tensor operators defined on maximally symmetric even dimensional spaces. We decompose the operators into their irreducible parts and obtain their corresponding eigenvalues. Using these eigenvalues, we are able to explicitly calculate $\zeta(0)$ for the cases of Euclidean spaces and $N$-spheres. In the $N$-sphere case, we make use of the Euler-Maclaurin formula to develop asymptotic expansions for the required sums. The resulting $\zeta(0)$ values for dimensions 2 to 10 are given in the Appendix.