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 $X^{23/24+10\varepsilon} \leq H \leq X^{1-\varepsilon}$, there exist constants $B_{f,h}$ such that $$ \sum_{X\leq n \leq 2X} f(n)f(n+h)-B_{f,h}X=O_{f,\varepsilon}\big(X^{1-\varepsilon^{2}/4}\big)$$ for all but $O_{f,\varepsilon}\big(HX^{-\varepsilon^{2}/3}\big)$ integers $h \in [1,H].$ 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 Matomaki, Radziwill and Tao. Also, we establish an upper bound for weighted exponential sums, which may be of independent interest.