A note on the Weyl formula for balls in ${\mathbb{R}}^d$

Let $\mathcal{B}=\{x\in {\mathbb{R}}^d:\mathrm{\mid }x\mathrm{\mid }<R\}$ ($d\ge 3$) be a ball. We consider the Dirichlet Laplacian associated with $\mathcal{B}$ and prove that its eigenvalue counting function has an asymptotics \begin{equation*} \mathscr{N}_\mathscr{B}(\mu)=C_d vol(\mathscr{B})\mu^d-C’_d vol(\partial \mathscr{B})\mu^{d-1}+O\left(\mu^{d-2+\frac{131}{208}}(\log \mu)^{\frac{18627}{8320}}\right) \end{equation*} as $\mu \to \mathrm{\infty }$.