  Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative
Ricci curvature, maximal volume growth and unique tangent cone at infinity. In
this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the
cross-section $X$. Denote by
$\alpha=\lim_{r\rightarrow\infty}\frac{\mathrm{Vol}(B_{r}(p))}{r^{n}}$ the
asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of
harmonic functions with polynomial growth of growth order at most $k$. In this
paper, we prove a upper bound of $h_{k}$ in terms of the counting function of
eigenvalues of $X$. As a corollary, we obtain
$\lim_{k\rightarrow\infty}k^{1-n}h_{k}=\frac{2\alpha}{(n-1)!\omega_{n}}$. These
results are sharp, as they recover the corresponding well-known properties of
$h_{k}(\mathbb{R}^{n})$. In particular, these results hold on manifolds with
nonnegative sectional curvature and maximal volume growth.
