On an estimate of Walfisz and Saltykov for an error term related to the Euler function
Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 203-236.

The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H\left(x\right)\ll {\left(logx\right)}^{2/3}{\left(loglogx\right)}^{4/3}$ for the error term $H\left(x\right)={\sum }_{n\le x}\frac{\phi \left(n\right)}{n}-\frac{6}{{\pi }^{2}}x$ 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 $H\left(x\right)\ll {\left(logx\right)}^{2/3}{\left(loglogx\right)}^{1+ϵ}$ 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 ${\left(\phi \left(n\right)/n\right)}^{r},{\left(\sigma \left(n\right)/n\right)}^{r}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}{\left(\sigma \left(n\right)/\phi \left(n\right)\right)}^{r}$ for every real value of $r$, and also to ${\sigma }^{\left(r\right)}\left(n\right)$, the sum of the exponential divisors $d$ of $n$ with ${p}^{\alpha }\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}¬\parallel d\phantom{\rule{4pt}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}{p}^{2\alpha }\parallel n\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\alpha >1$.

On étend la technique qui a permis à A. Walfisz d'établir (en 1962) l'estimation $H\left(x\right)\ll {\left(logx\right)}^{2/3}{\left(loglogx\right)}^{4/3}$ pour le terme d'erreur lié à la fonction d'Euler $H\left(x\right)={\sum }_{n\le x}\frac{\varphi \left(n\right)}{n}-\frac{6}{{\pi }^{2}}x,$ tout en incorporant à l'argument des simplifications rendues possibles par des travaux de A.I. Saltykov et de A.A. Karatsuba. On remarque en passant que la preuve proposée en 1960 par Saltykov de $H\left(x\right)\ll {\left(logx\right)}^{2/3}{\left(loglogx\right)}^{1+ϵ}$ contient une faute, qui une fois corrigée ne livre “que” le résultat de Walfisz. Les généralisations obtenues s'appliquent aux termes d'erreurs liés à diverses fonctions arithmétiques classiques, et moins classiques, comme par exemple à $\left({\varphi \left(n\right)/n\right)}^{r},{\left(\sigma \left(n\right)/n\right)}^{r}\right$ et ${\left(\sigma \left(n\right)/\sigma \left(n\right)\right)}^{r}$ pour chaque valeur réelle de $r$, ou encore à ${\sigma }^{\left(r\right)}\left(n\right)$, la somme des diviseurs exponentiels $d$ de $n$ tels que ${p}^{\alpha }\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}¬\parallel d\phantom{\rule{4pt}{0ex}}\text{si}\phantom{\rule{4pt}{0ex}}{p}^{2\alpha }\parallel n\phantom{\rule{4pt}{0ex}}\text{et}\phantom{\rule{4pt}{0ex}}\alpha >1$.

@article{JTNB_1998__10_1_203_0,
author = {Y.-F. S. P\'etermann},
title = {On an estimate of {Walfisz} and {Saltykov} for an error term related to the {Euler} function},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {203--236},
publisher = {Universit\'e Bordeaux I},
volume = {10},
number = {1},
year = {1998},
zbl = {0917.11047},
mrnumber = {1827292},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_1998__10_1_203_0/}
}
TY  - JOUR
AU  - Y.-F. S. Pétermann
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
SP  - 203
EP  - 236
VL  - 10
IS  - 1
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_1998__10_1_203_0/
LA  - en
ID  - JTNB_1998__10_1_203_0
ER  - 
%0 Journal Article
%A Y.-F. S. Pétermann
%T On an estimate of Walfisz and Saltykov for an error term related to the Euler function
%J Journal de théorie des nombres de Bordeaux
%D 1998
%P 203-236
%V 10
%N 1
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_1998__10_1_203_0/
%G en
%F JTNB_1998__10_1_203_0
Y.-F. S. Pétermann. On an estimate of Walfisz and Saltykov for an error term related to the Euler function. Journal de théorie des nombres de Bordeaux, Volume 10 (1998) no. 1, pp. 203-236. https://jtnb.centre-mersenne.org/item/JTNB_1998__10_1_203_0/

[1] U. Balakrishnan and Y.-F.S. Pétermann, The Dirichlet series of ζ(s)ζα (s + 1) f (s + 1): On an error term associated with its coefficients, Acta Arith. 75 (1996), 39-69. | Zbl

[2] A. Ivi, The Riemann zeta-function, John Wiley and Sons 1985. | MR | Zbl

[3] E. Grosswald. The average order of an arithmetical function, Duke Math. J. 23 (1956), 41-44. | MR | Zbl

[4] A.A. Karatsuba, Estimates for trigonometric sums by Vinogradov's method, and some applications, Proc. Steklov Inst. Math. (A.M.S English translation, 1973) 112 (1971), 251-265. | Zbl

[5] M.N. Korobov, Estimates of trigonometrical sums and their applications (in Russian), Uspekhi Mat. Nauk. 13 (4) (1958), 185-192. | MR | Zbl

[6] Y.-F.S. Pétermann and Jie Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175. | MR | Zbl

[7] A.I. Saltykov, On Euler's function (in Russian), Vestnik Moskovskogo Universiteta, Seriya I: Matematika, Mekhanika, no vol. number, fasc. number 6 (1960), 34-50. | MR | Zbl

[8] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres. Institut Elie Cartan 13 1990. | Zbl

[9] E.C. Titchmarsh, The theory of the Riemann zeta-function, Oxford, Clarendon Press 1951; second edition revised by D.R. Heath-Brown, ibid 1986. | MR

[10] A. Walfisz, Über die Wirksamkeit einiger Abschätzungen trigonometrischer Summen, Acta Arith. 4 (1958), 108-180. | MR | Zbl

[11] A. Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie. VEB Deutscher Verlag der Wissenschaften, Berlin 1963. | MR | Zbl