Generating function of the arithmetical function rd(n) and its relation to the Casimir energy

TL;DR

This paper derives analytical formulas for the generating function of the sum of d-squares arithmetical function, linking it to Casimir energy calculations in physics, and demonstrates their accuracy through numerical comparison.

Contribution

It provides a new analytical expression for the generating function of r_d(n) and connects it to physical Casimir energy in higher dimensions.

Findings

Derived a simplified formula with a single finite sum.

Numerical comparisons show negligible difference at small λ.

Linked the mathematical term to Casimir energy in physics.

Abstract

We obtain analytical expressions for the generating function $\xi_d(\lambda)$ of the sum of $d$-squares arithmetical function $r_d(n)$ where $\lambda$ is a free parameter. The original $d$-dimensional infinite sum is reduced to a formula containing a single finite sum over a convergent series. We compare the formulas to numerical computations and show that the percentage difference is negligible at small $\lambda$ for various values of $d$. $\xi_d(\lambda)$ divides naturally into two terms and we show that one term has a direct physical application to the $d$-dimensional Casimir energy of massless scalar fields in cubic cavities.