On $\sum _{n}\leq _{x}(\sigma ^{\ast } (n))$ and $\sum _{n}\leq _{x}(\phi ^{\ast } (n))$

Abstract:

Let ${\sigma ^ \ast }(n)$ and ${\varphi ^ \ast }(n)$ be the unitary analogues of $\sigma (n)$ and $\varphi (n)$ respectively. It is known that $E(x) = \sum \nolimits _{n \leqq x} {{\sigma ^ \ast }} (n) - ({\pi ^2}{x^2}/12\zeta (3)) = O(x{\log ^2}x)$ and \[ F(x) = \sum \limits _{n \leqq x} {{\varphi ^ \ast }(n) - \tfrac {1}{2}\alpha {x^2} = O(x{{\log }^2}x),} \] where $\alpha$ is a positive constant. In this paper we improve the order estimates of $E(x)$ and $F(x)$ to $E(x) = O(x{\log ^{5/3}}x)$ and \[ F(x) = O(x{\log ^{5/3}}x{(\log \log x)^{4/3}}).\]