  We calculate the triple correlations for the truncated divisor sum
$\lambda_{R}(n)$. The $\lambda_{R}(n)$'s behave over certain averages just as
the prime counting von Mangoldt function $\Lambda(n)$ does or is conjectured to
do. We also calculate the mixed (with a factor of $\Lambda(n)$) correlations.
The results for the moments up to the third degree, and therefore the
implications for the distribution of primes in short intervals, are the same as
those we obtained (in the first paper with this title) by using the simpler
approximation $\Lambda_{R}(n)$. However, when $\lambda_{R}(n)$ is used the
error in the singular series approximation is often much smaller than what
$\Lambda_{R}(n)$ allows. Assuming the Generalized Riemann Hypothesis for
Dirichlet $L$-functions, we obtain an $\Omega_{\pm}$-result for the variation
of the error term in the prime number theorem. Formerly, our knowledge under
GRH was restricted to $\Omega$-results for the absolute value of this
variation. An important ingredient in the last part of this work is a recent
result due to Montgomery and Soundararajan which makes it possible for us to
dispense with a large error term in the evaluation of a certain singular series
average. We believe that our results on $\lambda_{R}(n)$'s and
$\Lambda_{R}(n)$'s can be employed in diverse problems concerning primes.
