Additive Sums of Shifted Ternary Divisor Function

Fix a positive integer $X$ and multi-sets of complex numbers $\mathcal{I}$ and $\mathcal{J}$. We study the shifted convolution sum \[ D_{\mathcal{I},\mathcal{J}}(X,1) = \sum_{n\leq X} \tau_{\mathcal{I}}(n)\tau_{\mathcal{J}}(n+1), \] where $\tau_{\mathcal{I}}$ and $\tau_{\mathcal{J}}$ are shifted divisor functions. These sums naturally appear in the study of higher moments of the Riemann zeta function and additive problems in number theory. We review known results on $2k$-th moment of the Riemann zeta function and correlation sums associated with generalized divisor function. Assuming a conjectural bound on the averaged level of distribution of $\tau_{\mathcal{J}}(n)$ in arithmetic progressions, we present an asymptotic formula for $D_{\mathcal{I},\mathcal{J}}(X,1)$ with explicit main terms and power-saving error estimates.