On correlations of certain multiplicative functions

In this paper, we study sums of shifted products Sigma(n <= x) F(n)G(n-h) for any vertical bar h vertical bar <= 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 Sigma(n <= x) mu(2)(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. (C) 2016 Elsevier Inc. All rights reserved.