On asymptotics of shifted sums of Dirichlet convolutions

TL;DR

This paper derives asymptotic formulas for shifted sums of multiplicative functions related to Dirichlet convolutions, using the Hardy-Littlewood circle method and advanced cancellation techniques, under certain conjectural bounds.

Contribution

It provides new asymptotic results for shifted sums of multiplicative functions satisfying the Ramanujan conjecture, employing novel analytic techniques and bounds.

Findings

Asymptotic formulas hold for a wide range of H values.

Established bounds for weighted exponential sums.

Applied advanced circle method techniques to multiplicative functions.

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.