Geodesic
On the Riesz means of $\frac {n}{\phi (n)}$ – III
Ayyadurai Sankaranarayanan1 ; Saurabh Kumar Singh1
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.

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

Ayyadurai Sankaranarayanan 1 ; Saurabh Kumar Singh 1

1 School of Mathematics Tata Institute of Fundamental Research (TIFR) Homi Bhabha Road Mumbai 400 005, India 
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Ayyadurai Sankaranarayanan; Saurabh Kumar Singh. On the Riesz means of $\frac {n}{\phi (n)}$ – III. Acta Arithmetica, Tome 170 (2015) no. 3, pp. 275-286. doi : 10.4064/aa170-3-4. http://geodesic.mathdoc.fr/articles/10.4064/aa170-3-4/

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