A Sankaranarayanan ; Saurabh Kumar Singh - On the Riesz means of $\frac{n}{\phi(n)}$

hrj:179 - Hardy-Ramanujan Journal, January 1, 2013, Volume 36 - https://doi.org/10.46298/hrj.2013.179
On the Riesz means of $\frac{n}{\phi(n)}$

Authors: A Sankaranarayanan ; Saurabh Kumar Singh

Let $\phi(n)$ denote the Euler-totient function. We study the error term of the general $k$-th Riesz mean of the arithmetical function $\frac {n}{\phi(n)}$ for any positive integer $k \ge 1$, namely the error term $E_k(x)$ where \[ \frac{1}{k!}\sum_{n \leq x}\frac{n}{\phi(n)} \left( 1-\frac{n}{x} \right)^k = M_k(x) + E_k(x). \] The upper bound for $\left | E_k(x) \right |$ established here thus improves the earlier known upper bound when $k=1$.

Volume: Volume 36
Published on: January 1, 2013
Submitted on: March 3, 2015
Keywords: mean-value theorems, Riemann zeta-function, Generating functions,Euler-totient function,[MATH] Mathematics [math]


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