Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

We show that the expected asymptotic for the sums \∑X < n \≤ 2X\Λ(n) \Λ(n+h), \∑X < n \≤ 2X dₖ(n) dₗ(n+h), and \∑X< n \≤ 2X \Λ(n) dₖ(n+h) hold for almost all h \∈ [-H,H], provided X8/33+\ε \≤ H \≤ X1-\ε, with an error term on average an arbitrary power of the logarithm over the trivial bound. work of Mikawa, Perelli-Pintz and Baier-Browning-Marasingha-Zhao the range H \≥ X1/3+\ε. We also obtain an analogous for \∑ₙ \Λ(n) \Λ(N-n). Our proof uses the circle method and some oscillatory integral estimates(following a paper of Zhan) to reduce matters to establishing some mean-value for certain Dirichlet polynomials associated to "Type d₃" and"Type d₄" sums (as well as some other sums that are easier to treat). After H\\"older's inequality to the Type d₃ sum, one is left with two, one of which we can control using a short interval mean value of Jutila, and the other we can control using exponential sum estimates Robert and Sargos. The Type d₄ sum is treated similarly using the L² mean value theorem and the classical van der Corput exponential estimates.