The goal is the leading asymptotic for shifted correlations of divisor functions, the sum of d_k(n)d_l(n+h), for all k, l >= 1 and nonzero h. Before this work, no case with k, l >= 3 was known.

The proof replaces d_k by a density model and evaluates the model's correlation directly. The remainder is reduced to correlations where n and n+h have exactly one prime divisor in many prime 'banks'. Weighted-dilation graph estimates from prior work relate these fixed-shift correlations to averaged correlations with shifts hD1D2, which are then bounded by major/minor arc analysis using a zero-free region for Dirichlet L-functions. Lean 4 formalizations of the intermediate graph results borrowed from the prior work are available in a cited repository.

The sum of Φ(n/x)d_k(n)d_l(n+h) equals S_{k,l}(h)/((k-1)!(l-1)!) · x(log x)^{k+l-2} ∫Φ, with an error term saving a power (log x)^{-η}. The singular series S_{k,l}(h) is given explicitly by p-adic local integrals.