On asymptotics of shifted sums of Dirichlet convolutions

Abstract: The objective of this paper is to obtain asymptotic results for shifted sums of multiplicative functions of the form $g \ast 1$, where the function $g$ satisfies the Ramanujan conjecture and has conjectured upper bounds on square moments of its L-function. We establish that for $H$ within the range $X^{1-2\varepsilon} \leq H \leq X^{1-\varepsilon}$, there exist constants $B_{f,h}$ such that $$ \sum_{X\leq n \leq 2X} f(n)f(n+h)-B_{f,h}X=O_{f,\varepsilon}\big(X^{1-\varepsilon/2}\big)$$ for all but $O_{f,\varepsilon}\big(HX^{-1/44+6\varepsilon+o(1)}\big)$ integers $h \in [1,H].$ Additionally, we establish power-saving error terms for almost all integers $h$ in $[1, H]$ when $X^{87/88+7\varepsilon} \leq H \leq X^{1-\varepsilon}$ and apply them to deduce mean-value estimates of long Dirichlet polynomials. Our method is based on the Hardy-Littlewood circle method. In order to treat minor arcs, we use the convolution structure and a cancellation of $g(n)$ that are additively twisted, applying some arguments from a paper of Matom\"a{}ki, Radziwi\l{}\l{} and Tao.