On the Riesz means of $\frac {n}{\phi (n)}$ – III
Volume 170 / 2015
Acta Arithmetica 170 (2015), 275-286
MSC: Primary 11A25; Secondary 11N37.
DOI: 10.4064/aa170-3-4
Abstract
Let $\phi (n)$ denote the Euler totient function. We study the error term of the general $k$th Riesz mean of the arithmetical function ${n/\phi (n)}$ for any positive integer $k \ge 1$, namely the error term $E_k(x)$ where $$ \frac{1}{k!}\sum_{n \leq x}\frac{n}{\phi(n)} \left( 1-\frac{n}{x} \right)^{\!k} = M_k(x) + E_k(x). $$ For instance, the upper bound for $ | E_k(x) |$ established here improves the earlier known upper bounds for all integers $k$ satisfying $k\gg (\log x)^{1+\epsilon }$.
