 Oscillations d'un terme d'erreur lié à la fonction totient de Jordan
Journal de Théorie des Nombres de Bordeaux, Tome 3 (1991) no. 2, pp. 311-335.

Let ${J}_{k}\left(n\right):={n}^{k}{\prod }_{p\mid n}\left(1-{p}^{-k}\right)$ (the $k$-th Jordan totient function, and for $k=1$ the Euler phi function), and consider the associated error term ${E}_{k}\left(x\right):={\sum }_{n\le x}\phantom{\rule{4pt}{0ex}}{J}_{k}\left(n\right)-\frac{{x}^{k+1}}{\left(k+1\right)\zeta \left(k+1\right)}$. When $k\ge 2$, both ${i}_{k}:={E}_{k}\left(x\right){x}^{-k}$ and ${s}_{k}:=lim sup{E}_{k}\left(x\right){x}^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead ${I}_{k}:={lim inf}_{n\in ℕ,n\to \infty }{\sum }_{d\ge 1}\phantom{\rule{4pt}{0ex}}\frac{\mu \left(d\right)}{{d}^{k}}\left(\frac{1}{2}-\left\{\frac{n}{d}\right\}\right),$ since from [AS] ${i}_{k}={I}_{k}-{\left(\zeta \left(k+1\right)\right)}^{-}1$ and from the present paper ${s}_{k}=-{i}_{k}$. We show that ${I}_{k}$ belongs to an interval of the form $\left(\frac{1}{2\zeta \left(k\right)}-\frac{1}{\left(k-1\right){N}^{k-1}},\frac{1}{2\zeta \left(k\right)}\right)$, where $N=N\left(k\right)\to \infty$ as $k\to \infty$. From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of ${I}_{k}$. We apply this algorithm to the small values of $k$ and obtain $.29783 and $.46196896<{I}_{4}<.46196916$.

@article{JTNB_1991__3_2_311_0,
author = {P\'etermann, Y.-F. S.},
title = {Oscillations d'un terme d'erreur li\'e \a la fonction totient de {Jordan}},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {311--335},
publisher = {Universit\'e Bordeaux I},
volume = {3},
number = {2},
year = {1991},
doi = {10.5802/jtnb.53},
zbl = {0749.11041},
mrnumber = {1149800},
language = {fr},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.53/}
}
Y.-F. S. Pétermann. Oscillations d'un terme d'erreur lié à la fonction totient de Jordan. Journal de Théorie des Nombres de Bordeaux, Tome 3 (1991) no. 2, pp. 311-335. doi : 10.5802/jtnb.53. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.53/`

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