Existence of all the asymptotic $\lambda$-th means for certain arithmetical convolutions

Abstract

Let $E$ designate either of the classical error terms for the summatory functions of the arithmetical functions $\phi(n)/n$ and $\sigma(n)/n$ ($\phi$ is Euler's function and $\sigma$ the divisor function). By following an idea of Codecà's [3] and by refining some of his estimates we prove that $|E|$ has asymptotic $\lambda$-th order means for all positive real numbers $\lambda$. We also prove that $E$ has asymptotic $k$-th order means for all positive integers $k$, and that this mean is zero whenever $k$ is odd. The results obtained can be applied to functions other than $E$ as well, such as the functions $P$ and $Q$ of Hardy and Littlewood[8], or the divisor functions $G_{-1,k}$[9].