Estimates of convolutions of certain number-theoretic error terms

Several estimates for the convolution function C[f (x)] := x 1 f (y)f (x/y)(dy/y) and its iterates are obtained when f (x) is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for T 0 |ζ(1/2 + it)| 2k dt (k = 1, 2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin-Selberg convolution. 1. Convolution functions and Mellin transforms. Let ᏸ denote the set of functions f (x) ∈ L 1 (1, ∞) for which there exists a constant α f 0 such that f (x) ε x α f +ε. Actually it is more precise to define α f as the infimum of the constants for which (1.1) holds. Here and later ε > 0 denotes arbitrarily small constants, not necessarily the same ones at each occurrence. The notation A ε B (the same as A = O ε (B)) means that |A| C(ε)B for some positive constant C(ε), which depends only on ε. We define the convolution of functions f ,g ∈ ᏸ as f g (x) := x 1 f (y)g x y dy y. (1.2) The operation is commutative, associative, and distributive with respect to ordinary addition. Note that f g ∈ ᏸ since it obviously lies in L 1 (1, ∞) and, moreover, x 1 f (y)g x y dy y ε x 1 y α f +ε x y αg +ε dy y ε x γ+ε (1.3) with γ = γ(f , g) = max(α f ,α g). We also set, if f ∈ ᏸ, C f (x) := f f (x) = x 1 f (y)f x y dy y (x 1). (1.4) More generally, the iterates of C[f (x)] are defined as C 1 f (x) ≡ C f (x) , C k f (x) := C C k−1 f (x) (x 1,k 2).