Convolutions and mean square estimates of certain number-theoretic error terms

Abstract

We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for $f(x)$. Some applications are given, in particular to $|\zeta(1/2+ix)|^{2k}$ and the classical Rankin--Selberg problem from analytic number theory.