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 $X^{23/24+10\epsilon }\le H\le X^{1-\epsilon }$, there exist constants $B_{f,h}$ such that $$\sum _{X\le n\le 2X}f(n)f(n+h)-B_{f,h}X=O_{f,\epsilon }\left(X^{1-{\epsilon }^2/4}\right)$$ for all but $O_{f,\epsilon }\left(H{X}^{-{\epsilon }^2/3}\right)$ 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.

Fixed minor errors, added Lemma 3.1, and improved the main result