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.