On a polynomial zeta function

TL;DR

This paper introduces a polynomial zeta function linked to mathematical physics, computes its values at zero, and applies these results to determine the determinant of the Dirac operator on quaternionic vector spaces.

Contribution

It presents a new polynomial zeta function and a simple method to evaluate it at zero, with applications to physics and operator determinants.

Findings

Computed the polynomial zeta function at s=0 and its derivative

Determined the determinant of the Dirac operator on quaternionic spaces

Provided a simple technique for evaluating related zeta functions

Abstract

We introduce a polynomial zeta function $\zeta^{(p)}_{P_n}$, related to certain problems of mathematical physics, and compute its value and the value of its first derivative at the origin $s=0$, by means of a very simple technique. As an application, we compute the determinant of the Dirac operator on quaternionic vector spaces.