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Published online by Cambridge University Press: 17 May 2019
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 The value of 
$$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$
$\unicode[STIX]{x1D700}(m)$ 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.
The author is funded by a Departmental Award and by an EPSRC Doctoral Training Partnership Award.
