On the Riesz means of $\frac {n}{\phi (n)}$ – III

Ayyadurai Sankaranarayanan ¹   ; Saurabh Kumar Singh ¹

¹ School of Mathematics Tata Institute of Fundamental Research (TIFR) Homi Bhabha Road Mumbai 400 005, India

Acta Arithmetica, Tome 170 (2015) no. 3, pp. 275-286

Résumé

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 $k\gg (\log x)^{1+\epsilon }$.

Détail

DOI : 10.4064/aa170-3-4

Keywords: phi denote euler totient function study error term general kth riesz mean arithmetical function phi positive integer namely error term where frac sum leq frac phi frac right instance upper bound established here improves earlier known upper bounds integers satisfying log epsilon