Order of zeta functions for compact even-dimensional symmetric spaces

TL;DR

This paper investigates zeta functions associated with vector bundles over compact even-dimensional symmetric spaces, showing they can be expressed as entire functions with order bounded by the space's dimension.

Contribution

It establishes a bound on the order of zeta functions in terms of the dimension of the symmetric space, linking geometric properties to analytic behavior.

Findings

Zeta functions can be expressed as entire functions.

The order of these functions is at most the dimension of the space.

Results apply to compact, even-dimensional, locally symmetric spaces.

Abstract

Some zeta functions which are naturally attached to the locally homogeneous vector bundles over compact locally symmetric spaces of rank one are investigated. We prove that such functions can be expressed in terms of entire functions whose order is not larger than the dimension of the corresponding compact, even-dimensional, locally symmetric space.