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

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.

18 pages, 2 figures