On an estimate of Walfisz and Saltykov for an error term related to the Euler function

@article{Pétermann1998,
abstract = {The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H(x) \ll (\log x)^\{2/3\} (\log \log x)^\{4/3\}$ for the error term $H(x) = \sum _\{n \le x\} \frac\{\phi (n)\}\{n\} - \frac\{6\}\{\pi ^2\}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H(x) \ll (\log x)^\{2/3\}(\log \log x)^\{1 + \epsilon \}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical functions, as for instance to $(\phi (n)/n)^r, (\sigma (n)/n)^r \text\{ and \} (\sigma (n)/ \phi (n))^r$ for every real value of $r$, and also to $\sigma ^\{(r)\}(n)$, the sum of the exponential divisors $d$ of $n$ with $p^\alpha \!\!\! \lnot \{\parallel \} d \text\{ if \} p^\{2\alpha \} \Vert n \text\{ and \} \alpha > 1$.},
author = {Pétermann, Y.-F. S.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {asymptotic formulae for the Euler function with remainder term; sum of divisors; estimation of exponential sums with primes; multiplicative arithmetical functions; Walfisz method},
language = {eng},
number = {1},
pages = {203-236},
publisher = {Université Bordeaux I},
title = {On an estimate of Walfisz and Saltykov for an error term related to the Euler function},
url = {http://eudml.org/doc/248159},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Pétermann, Y.-F. S.
TI - On an estimate of Walfisz and Saltykov for an error term related to the Euler function
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 1
SP - 203
EP - 236
AB - The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H(x) \ll (\log x)^{2/3} (\log \log x)^{4/3}$ for the error term $H(x) = \sum _{n \le x} \frac{\phi (n)}{n} - \frac{6}{\pi ^2}x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H(x) \ll (\log x)^{2/3}(\log \log x)^{1 + \epsilon }$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical functions, as for instance to $(\phi (n)/n)^r, (\sigma (n)/n)^r \text{ and } (\sigma (n)/ \phi (n))^r$ for every real value of $r$, and also to $\sigma ^{(r)}(n)$, the sum of the exponential divisors $d$ of $n$ with $p^\alpha \!\!\! \lnot {\parallel } d \text{ if } p^{2\alpha } \Vert n \text{ and } \alpha > 1$.
LA - eng
KW - asymptotic formulae for the Euler function with remainder term; sum of divisors; estimation of exponential sums with primes; multiplicative arithmetical functions; Walfisz method
UR - http://eudml.org/doc/248159
ER -