The distribution of 4-full numbers

Let Q(x) denote the number of 4-full numbers not exceeding x. It is well known that $$Q(x) = sumlimits_{j = 4}^7 {r_j x^{1/j} + R(x)}$$ where $$r_j = mathop {res}limits_{s = 1/j} (F(s)/s), F(s) = mathop prod limits_P left( {1 + frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} right)$$ and R(x) is the remainder. This paper proves that $$R(x) ll x^{3626/35461 + varepsilon }$$ where ε is any positive number.