Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem

Abstract

We calculate the triple correlations for the truncated divisor sum $λ_{R}(n)$. The $λ_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $Λ(n)$ does or is conjectured to do. We also calculate the mixed (with a factor of $Λ(n)$) 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 $Λ_{R}(n)$. However, when $λ_{R}(n)$ is used the error in the singular series approximation is often much smaller than what $Λ_{R}(n)$ allows. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain an $Ω_{\pm}$-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to $Ω$-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 $λ_{R}(n)$'s and $Λ_{R}(n)$'s can be employed in diverse problems concerning primes.