Spectral sum rules on a $d$--sphere

We derive spectral sum rules for inverse powers of the eigenvalues of the Helmholtz equation on a $d$-sphere in the presence of an arbitrary density. By adopting a rigorous renormalization scheme, we remove the divergent contributions of the zero mode and obtain exact expressions for the sum rules without requiring an explicit determination of the eigenvalues, which is generally impossible. As an application, we derive explicit sum rules for the density $\Sigma(\Omega) = 1 + \kappa Y_{1,\vec{0}}(\Omega)$ in $d=3,4,5$ dimensions and compare them with numerical estimates obtained by approximating the low-lying part of the spectrum with the Rayleigh--Ritz method and the high-energy part with Weyl's formula.