Additive Sums of Generalized Shifted Ternary Divisor Functions

Source abstract

In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum $D_{\mathcal{I},\mathcal{J}}(X,1)=\sum_{n \leq X} τ_{\mathcal{I}}(n) τ_{\mathcal{J}}(n + 1)$, where $\mathcal{I}=\{α_1,α_2,α_3\}$ and $\mathcal{J} = \{β_1, β_2, β_3\}$, with $|α_i|, |β_i|\leq δ$ for $i=1,2,3$, and $0<δ<1/44$. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function $τ_{\mathcal{J}}$ in arithmetic progressions up to the level $2/3$.