The asymptotics for the shifted convolutions of a class of multiplicative functions

Abstract

Let ${\lambda }_F\left(n\right)$ be a multiplicative function and its Dirichlet series $F\left(s\right)={\sum }_{n=1}^{\infty }\frac{{\lambda }_F\left(n\right)}{n^s}$ satisfies three reasonable conditions. We establish an asymptotic formula of shifted sum of the form$\sum _{X\le n\le 2X}{\lambda }_F\left(n\right)\overline{{\lambda }_F(n+h)}$for almost all $h\in [1,H]$, provided that $X^{s_1+\epsilon }\le H\le X^{1-\epsilon }$, where $0\le s_1<1$ is a constant that is only related to $F$. Let ${\left\{{\lambda }_f\right(n\left)\right\}}_{n\ge 1}$ be normalized Hecke eigenvalues of a holomorphic cusp form f. As applications, we consider a few special cases, including $d_k\left(n\right)⁎{\lambda }_{i,f}\left(n\right)$, $d_k\left(n\right){\lambda }_{i,f}\left(n\right)$, and $d_k\left(n\right){\lambda }_f^2\left(n\right)$, where ${\lambda }_{i,f}={\lambda }_f⁎\cdots ⁎{\lambda }_f$ is the i-fold convolution of ${\lambda }_f$, $d_k$ is the kth divisor function. Further, under a mild hypothesis, we see that ${\lambda }_f^{\ell }\left(n\right)$ is also within our range of results.