  The notion of spectral dimension was introduced by Chakraborty and Pal in
\cite{cp}. In this paper, we show that the spectral dimension of the ring of
$p$-adic integers, $\mathbb{Z}_p$, is equal to its manifold dimension, which is
$0$. Finally, we determine the $K$-groups of $\mathbb{Z}_p$, and show that the
generators of $K_0(\mathbb{Z}_p)$ can be expressed as finite span of the
characters of $\mathbb{Z}_p$.
