Justification for zeta function regularization

TL;DR

This paper justifies using zeta function regularization in physical problems involving divergent sums, demonstrating its application in graphene magnetization and Casimir force calculations.

Contribution

It provides a theoretical justification for the use of zeta functions with negative variables in regularizing divergent sums in physics.

Findings

Zeta functions with negative variables naturally arise from physical cut-offs.

Application to graphene magnetization shows the method's relevance.

Justification extends to Casimir force calculations at zero temperature.

Abstract

Using the fact that a finite sum of power series are given by the difference between two zeta functions, we justify the usage of the zeta function with a negative variable in physical problems to avoid the divergence of the infinite sum. We will show that in the case of magnetization of graphene, the zeta function with negative variable arises as a result of cut-off energy between two consecutive Landau levels. Furthermore, similar justification can be applied to the case of zero temperature Casimir force in parallel-plate geometry.