  This paper explores the generation of primordial black holes (PBHs) and
scalar-induced gravitational waves (SIGWs) from the inflation potential with a
tiny Lorentz function bump. We choose the Starobinsky model as basic potential,
which satisfies the condition of observational constraints of the CMB at a
large scale. We find that the tiny Lorentz function bump can enhance the
primordial curvature power spectrum to $\mathcal{O}(10^{-2})$ at a small scale,
leading to the formation of PBHs with sufficient abundance. Furthermore, we
discover that the abundance of PBHs with mass $10^{-12}M_\odot$ is
approximately $1$, which can make up almost all dark matter. In addition, the
SIGWs generated by our models can be tested by the Square Kilometre Array
(SKA), Pulsar Timing Arrays (PTA), TianQin, Taiji, Laser Interferometer Space
Antenna (LISA), and DECIGO. It should be emphasized that, compared with the
tiny Gaussian bump case, the Lorentz function bump case can easily produce a
sufficient abundance of PBHs with a wider mass range, and accompanying a
broader frequency range of SIGWs, which predicts the advantage of the Lorentz
function bump for basic potential with big e-folding number $N$ around
$55\sim60$.
