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

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 $\frac {n}{\phi(n)}$ 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 $k=1$.
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A Sankaranarayanan, Saurabh Kumar Singh. On the Riesz means of $\frac{n}{\phi(n)}$. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2013, 36, pp.8 - 20. ⟨hal-01112687⟩

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