Partition function on spheres: how (not) to use zeta function regularization

TL;DR

This paper explores the use of zeta function regularization on spheres, analyzing when formal manipulations yield correct results despite its non-linearity, through various examples.

Contribution

It clarifies the conditions under which zeta function regularization provides correct outcomes despite its inherent non-linearity.

Findings

Formal manipulations can be valid in specific cases

Zeta function regularization is generally non-linear

Examples demonstrate when the method works or fails

Abstract

It is known that not all summation methods are linear and stable. Zeta function regularization is in general non-linear. However, in some cases formal manipulations with "zeta function" regularization (assuming linearity of sums) lead to correct results. We consider several examples and show why this happens.