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

Pétermann, Y.-F. S.

Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 1, pp. 203-236.

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 $\left({\varphi \left(n\right)/n)}^r,{(\sigma \left(n\right)/n)}^r\right.$ 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 $p^{\alpha }\neg \parallel d\text{si}{p}^{2\alpha }\parallel n\text{et}\alpha >1$.

The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{4/3}$ 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 $H\left(x\right)\ll {(logx)}^{2/3}{(loglogx)}^{1+ϵ}$ 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 ${(\phi \left(n\right)/n)}^r,{(\sigma \left(n\right)/n)}^r\text{and}{(\sigma \left(n\right)/\phi \left(n\right))}^r$ 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 $p^{\alpha }\neg \parallel d\text{if}{p}^{2\alpha }\parallel n\text{and}\alpha >1$.

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