On correlations of certain multiplicative functions

In this paper, we study sums of shifted products ∑n≤xF(n)G(n−h) for any |h|≤x/2 and arithmetic functions F=f⁎1 and G=g⁎1, 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 ∑n≤xμ2(n)G(n−h), where G=g⁎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.