Higher correlations of divisor sums related to primes II: Variations of the error term in the prime number theorem

We calculate the triple correlations for the truncated divisor sum lambda(R)(n). The lambda(R)(n) 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 (GRH) for Dirichlet L-functions, we obtain an Omega(+/-)-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 the sums lambda(R)(n) and Lambda(R)(n) can be employed in diverse problems concerning primes.