We prove that for every
$m\geq 0$
there exists an
$\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$
such that if
$0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$
and
$x$
is sufficiently large in terms of
$m$
and
$\unicode[STIX]{x1D706}$
, then
$$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$
The value of
$\unicode[STIX]{x1D700}(m)$
and the dependence of the implicit constant on
$\unicode[STIX]{x1D706}$
and
$m$
may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters
$m$
and
$\unicode[STIX]{x1D706}$
to vary as functions of
$x$
or replacing the set
$\mathbb{P}$
of all primes by primes belonging to certain specific subsets.