The Conrey–Gonek–Ivić additive divisor sum conjecture

Let $d_k(n)$ denote the $k$-fold divisor function, and define the additive divisor correlation $D_{h,k,l}(x)$ as in the paper. For fixed $k,l\in\mathbb{N}$ and $h=O(x^{1-\epsilon})$ for every fixed $\epsilon>0$, there should exist a number $\delta>0$ and a polynomial $P_{h,k,l}$ of degree $k+l-2$ such that

$$D_{h,k,l}(x)=xP_{h,k,l}(\log x)+O_{\epsilon,k,l}(x^{1-\delta}).$$

Moreover, with $h=\prod p^{\gamma}$, the leading term is conjectured to satisfy

$$\frac{D_{h,k,l}(x)}{x\log^{k+l-2}x}\sim\frac{C_{k,l}f_{k,l}(h)}{(k-1)!(l-1)!}$$

as $x\to\infty$, where

$$C_{k,l}=\prod_p\left(\left(1-p^{-1}\right)^{l-1}+\left(1-p^{-1}\right)^{k-1}-\left(1-p^{-1}\right)^{k+l-2}\right)$$

and

$$f_{k,l}(h)=\prod_{p\mid h}\frac{(1-p^{-1})\sum_{\alpha=0}^{\gamma}d_{l-1}(p^{\alpha})\sum_{\beta=\alpha}^{\infty}d_k(p^{\beta})p^{-\beta}+d_k(p^{\gamma})\sum_{\alpha=\gamma+1}^{\infty}d_{l-1}(p^{\alpha})p^{-\alpha}}{(1-p^{-1})^{1-k}+(1-p^{-1})^{1-l}-1}.$$

This is the central conjecture on additive divisor correlations, extending the known asymptotic results for cases such as $l=2$ to the difficult range $k,l\geq 3$.