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

Volume 170 / 2015

Ayyadurai Sankaranarayanan, Saurabh Kumar Singh Acta Arithmetica 170 (2015), 275-286 MSC: Primary 11A25; Secondary 11N37. DOI: 10.4064/aa170-3-4

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

Authors

  • Ayyadurai SankaranarayananSchool of Mathematics
    Tata Institute of Fundamental Research (TIFR)
    Homi Bhabha Road
    Mumbai 400 005, India
    e-mail
  • Saurabh Kumar SinghSchool of Mathematics
    Tata Institute of Fundamental Research (TIFR)
    Homi Bhabha Road
    Mumbai 400 005, India
    e-mail

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