On correlations of certain multiplicative functions

In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$forany$|h| \leq x/2$andarithmeticfunctions$F=f1$and$G=g1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of magnitude of $f$ and $g$. We also provide asymptotic formula for sums of the type $\sum _{n\le x}{\mu }^2(n)G(n-h)$, where $G=g\ast 1$ and $g$ is small. For small order of magnitudes of $f$ and $g$, we improve the error terms and make them independent of $h$.