Sign changes of error terms related to arithmetical functions

Let H(x) = ∑ n≤x φ(n) n − 6 π2x. Motivated by a conjecture of Erdös, Lau developed a new method and proved that #{n ≤ T : H(n)H(n + 1) < 0} T. We consider arithmetical functions f(n) = ∑ d|n bd d whose summation can be expressed as ∑ n≤x f(n) = αx+P (log(x))+E(x), where P (x) is a polynomial, E(x) = − ∑ n≤y(x) bn n ψ ( x n ) + o(1) and ψ(x) = x− bxc − 1/2. We generalize Lau’s method and prove results about the number of sign changes for these error terms.