On the sums Σn ⩽ x A(f(n)) and Σp ⩽ x A(f(p))

For a complex number s and an arithmetical function α, we write A(n) = Σdδ = nα(d) δs and A∗(n) = A(n) n−s. In this paper, we prove asymptotic formulae for the sums Σn⩽x A(f(n)), Σp⩽x A(f(p)), Σn⩽x A∗(f(n)), and Σp⩽x A∗(f(p)), where Re s > 0, f is a nonconstant polynomial with integer coefficients, and α satisfies a growth condition. Several illustrations are given which incidentally refine results due to R. Bellman, W. Schwarz, and W. A. Webb.