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

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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 \({| E_k(x)|}\) established here thus improves the earlier known upper bounds for all integers \({k\geq 1}\).

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

    Ayyadurai Sankaranarayanan & Saurabh Kumar Singh

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  1. Ayyadurai Sankaranarayanan
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  2. Saurabh Kumar Singh
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Correspondence to Ayyadurai Sankaranarayanan.

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Sankaranarayanan, A., Singh, S.K. On the Riesz means of \({\frac{n}{\phi(n)}}\)-II. Arch. Math. 103, 329–343 (2014). https://doi.org/10.1007/s00013-014-0691-8

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