  We study the integrated density of states (IDS) of a discrete Schr\"odinger
operator $H=-\Delta+V$ in ${\mathbb Z}^d$. We introduce a box counting
function, defined through the discrete landscape function of $H$. For any
deterministic bounded potential, we give estimates for the IDS from above and
below by the landscape box counting function. For the Anderson model, we get a
refined lower bound for the IDS, throughout the spectrum. In particular, near
the bottom of the spectrum, we show that both the IDS and the box counting
function can be estimated by the common distribution of the random potential,
which yields an improved bound on the so-called Lifschitz tails for disordered
systems. The work extends the recent landscape law on ${\mathbb R}^d$ in [DFM]
to tight-binding models on ${\mathbb Z}^d$.
